A negative number line: Negative numbers on the number line (practice)

Posted on 5 июля, 2020 by alexxlab

\circ F[/latex], which is [latex]20[/latex] degrees below [latex]0[/latex].

Temperatures below zero are described by negative numbers.

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.

Elevations can also be represented by negative numbers. The elevation at sea level is [latex]0[/latex] feet. Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about [latex]1,302[/latex] feet below sea level, so the elevation of the Dead Sea can be represented as [latex]-1,302[/latex] feet. Refer to the image below for a depiction.

The surface of the Mediterranean Sea has an elevation of [latex]0[/latex] ft. The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation.

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of [latex]500[/latex] feet. Its position would then be [latex]-500[/latex] feet as labeled in the image below.

Depths below sea level are described by negative numbers. A submarine [latex]500[/latex] ft below sea level is at [latex]-500[/latex] ft.

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at [latex]0[/latex] and showed the counting numbers increasing to the right as shown in the number line below. The counting numbers [latex](1, 2, 3, \ldots )[/latex] on the number line are all positive. We could write a plus sign, [latex]+[/latex], before a positive number such as [latex]+2[/latex] or [latex]+3[/latex], but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with [latex]-1[/latex] at the first mark to the left of [latex]0,-2[/latex] at the next mark, and so on. Refer to the number line below for reference.

On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

example

Plot the numbers on a number line:

[latex]3[/latex]
[latex]-3[/latex]
[latex]-2[/latex]

Solution
Draw a number line. Mark [latex]0[/latex] in the center and label several units to the left and right.

1. To plot [latex]3[/latex], start at [latex]0[/latex] and count three units to the right. Place a point as shown in the number line below.

2. To plot [latex]-3[/latex], start at [latex]0[/latex] and count three units to the left. Place a point as shown in the number line below.

3. To plot [latex]-2[/latex], start at [latex]0[/latex] and count two units to the left. Place a point as shown in the number line below.

try it

In the video below we show more examples of how to plot integers on a number line.

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See the number line below.

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation [latex]a<b[/latex] (read [latex]a[/latex] is less than [latex]b[/latex] ) when [latex]a[/latex] is to the left of [latex]b[/latex] on the number line. We write [latex]a>b[/latex] (read [latex]a[/latex] is greater than [latex]b[/latex] ) when [latex]a[/latex] is to the right of [latex]b[/latex] on the number line. This is shown for the numbers [latex]3[/latex] and [latex]5[/latex] in the image below.

The number [latex]3[/latex] is to the left of [latex]5[/latex] on the number line. So [latex]3[/latex] is less than [latex]5[/latex], and [latex]5[/latex] is greater than [latex]3[/latex].

The numbers lines to follow show a few more examples.

[latex]4[/latex] is to the right of [latex]1[/latex] on the number line, so [latex]4>1[/latex].
[latex]1[/latex] is to the left of [latex]4[/latex] on the number line, so [latex]1<4[/latex].

[latex]-2[/latex] is to the left of [latex]1[/latex] on the number line, so [latex]-2<1[/latex].
[latex]1[/latex] is to the right of [latex]-2[/latex] on the number line, so [latex]1>-2[/latex].

[latex]-1[/latex] is to the right of [latex]-3[/latex] on the number line, so [latex]-1>-3[/latex].
[latex]-3[/latex] is to the left of [latex]-1[/latex] on the number line, so [latex]-3<-1[/latex].

example

Order each of the following pairs of numbers using [latex]&lt[/latex]; or [latex]\text{>:}[/latex]

[latex]14[/latex] and [latex]6[/latex]
[latex]-1[/latex] and [latex]9[/latex]
[latex]-1[/latex] and [latex]- 4[/latex]
[latex]2[/latex] and [latex]- 20[/latex]

Show Solution

try it

In the video below we show more examples of how to use inequality symbols to compare integers.

Basic Rules for Positive and Negative Numbers

Numbers higher than zero are called positive numbers, and numbers lower than zero are negative numbers. That means they fall at either side of the number line. However, just because they’re on the same line doesn’t mean they follow the same rules! Keep reading for a list of the basic rules for using positive and negative numbers in math.

rules for adding and subtracting two numbers positive and negative

Rules for Signed Numbers

When using positive and negative numbers, you use the rules for signed numbers (numbers with positive or negative signs in front of them). Also known as operations for signed numbers, these steps can help you avoid confusion and solve math problems as quickly — and correctly — as possible.

Follow these rules to determine the best way to add, subtract, multiply, and divide positive and negative numbers. Remember, if there is no + or — sign, the number is positive.

Addition: Same Signs, Add the Numbers

When you’re adding two numbers together and they have the same sign (two positive or two negative numbers), add the numbers and keep the sign. For example:

1 + 1 = 2
51 + 32 = 83
-14 + (-6) = -20
-196 + (-71) = -267

Notice that equations with two positive numbers have positive sums, and equations with two negative numbers have negative sums. If you’re using a number line to solve the problem, adding two positive numbers will go farther to the positive side, and adding two negative numbers will go farther on the negative side.

Addition: Different Signs, Subtract the Numbers

If you’re adding positive and negative numbers together, subtract the smaller number from the larger one and use the sign from the larger number. For example:

6 + (-5) = 1
-17 + 22 = 5
-100 + 54 = -45
299 + (-1) = 298

As you can see, adding numbers with different signs is really a form of subtraction. When using a number line, your sum will end up closer to zero.

Subtraction: Switch to Addition

Subtracting positive and negative numbers means that you add the opposite numbers, or additive inverse. Change the subtraction sign to addition and change the sign that follows to its opposite. Then follow the steps for addition. For example:

-3 — (+5) becomes -3 + (-5) = -8
9 — (-7) becomes 9 + (+7) = 16
-14 — (+8) becomes -14 + (-8) = -22
25 — (-90) becomes 25 + (+90) = 115

A good tip is that whenever you see a negative sign and a minus sign together, such as in 9 — (-7), immediately make them positive signs. The negative signs cancel each other out, and the equation becomes an addition problem.

Multiplication and Division: Same Sign, Positive Result

It seems like multiplication and division would be more complicated than addition and subtraction, but they’re actually much simpler. The rule for multiplying positive and negative numbers with the same sign (two positive or two negative) is that the product will always be positive. For example:

8 x 4 = 32
(-8) x (-4) = 32
10 x 9 = 90
(-10) x (-9) = 90

The same rule applies for division. When dividing a number by another number with the same sign, the quotient (answer) is positive. For example:

12 ÷ 6 = 2
-12 ÷ (-6) = 2
100 ÷ 5 = 20
-100 ÷ (-5) = 20

Why does multiplying or dividing two negative numbers always equal a positive number? Like subtracting negative numbers, these operations turn the negatives into their opposite (inverse). You are essentially subtracting the negative number several times — and as seen above, subtracting negative numbers results in a positive equation.

Multiplication and Division: Opposite Sign, Negative Result

When multiplying a positive and a negative, the product will always be negative. It doesn’t matter what order the signs are in. For example:

6 x (-7) = -42
-7 x 6 = -42
12 x (-11) = -132
-11 x 12 = -132

In all of these cases, you first need to multiply or divide the numbers. Then decide whether the product or quotient is positive (two positives or two negatives in the equation) or negative (one positive and one negative in the equation).

Like and Unlike Signs in Addition and Subtraction

Another way to think about adding positive and negative numbers is to look at the signs in a row. Two like signs in a row (++ or —) mean you add the numbers, while two unlike signs in a row (+- or -+) mean that you subtract. For example:

7 + (+2) = 9 (++ are like signs, so the equation is addition)
9 + (-8) = 1 (+- are unlike signs, so the equation is subtraction)
11 — (+13) = 2 (-+ are unlike signs, so the equation is subtraction)
15 — (-10) = 25 (— are like signs, so the equation is addition)

This method follows the same rules as above but might help you solve the problem more quickly if you prefer to work out the signs beforehand. Once you understand positive and negative numbers conceptually, you can decide which method works best for you.

Understanding Math Foundations

Once you know the basics of math and its rules, the entire mathematical world is open to you. Unlike other subjects, math isn’t nuanced or up for interpretation — it just is what it is! For more math practice, check out the steps to long division problems (with examples). You can also review the different types of numbers in math before your next math assignment.

Staff Writer

3 Free Negative Number Line PDFs + Worksheets

This post may contain affiliate links or sponsored content, read our Disclosure Policy.

These three negative number line PDFs are a great visual tool to introduce your kid to negative numbers and how they relate to positives. Put that knowledge to the test with the two number line with negatives worksheets, which cover addition and subtraction. Oh, and did I mention all five downloads are totally free? 🙂

Understanding negative numbers, not to mention how they compare or work with positive numbers, can be pretty challenging for kids. That’s where number lines come in handy. Number lines provide a visual of the relationship and can help kids learn to move between negative and positive numbers whether adding, subtracting, or multiplying.

Below are three number line with negatives PDFs that will take your kid from a small range of negative and positive numbers to a much larger range. Your child can then practice adding and subtracting negative and positive numbers with the number line negative numbers worksheets. (Don’t worry, there’s an answer key for the negative number lines worksheets so you can easily check your kid’s progress.)

Negative Number Line: -10 to 10

Get started with this number line with negative numbers going from -10 to 10. This is a great beginner negative number line because the range of numbers is relatively small.

Number Line with Negatives: -20 to 20

Expand on the number line negative numbers with this option that goes from -20 to 20. Hopefully this printable will help your kid start to understand that any positive number can also be a negative number. In other words, as high as they can count—50, 100, 1,000—there can also be a negative.

Number Line Negative Numbers: -100 to 100

When your child is ready to work with large numbers, download this negative number lines PDF. It goes from -100 all the way up to 100.

You can turn this number line with negative numbers into more of a worksheet by blacking out certain numbers and having your kid fill them in. You can also write math problems next to the number line.

Negative Number Lines Worksheet: Addition

When your child is ready to start putting what he or she has learned into practice, download this number line with negatives worksheet. It focuses solely on addition problems.

If you want to get more mileage out of this number line with negative numbers worksheet, once your kid can correctly solve all the addition problems, go through and replace the plus signs with minus signs, turning them into subtraction problems.

Negative Number Line Worksheet: Addition Answer Key

Number Line with Negative Numbers Worksheet: Subtraction

This negative number line worksheet allows your kid to practice subtracting negative and positive numbers. As with the addition worksheet, you can get more out of this worksheet by replacing the minus signs with plus signs, taking it from a subtraction to an addition worksheet.

Negative Number Line Worksheet: Subtraction Answer Key

Bonus: Another thing you may want to work on is negative numbers with decimals. You can create your own negative number line with decimals by using these free blank number lines (printable PDF downloads). In addition to creating a negative number line with decimals, you can also use these blank printables to introduce negative fractions.

More Number Line Printables and Worksheets

9 Free Printable Number Lines For Kids

4 Free Printable Fractions Number Lines (PDF Downloads) + Worksheet

3 Free Number Line to 100 Printables (PDF Downloads)

8 Free Printable 4th and 5th Grade Decimal Number Lines

Free Printable Number Line to 20 PDF Downloads + Worksheet

Free Printable Number Line to 10 Worksheet + 3 Number Line PDF Downloads

7 Free Integer Number Line Printables + 3 Adding Integers Number Lines
Negative and Positive Number Line

Free Printable Graphing Inequalities on a Number Line Practice Worksheet + 2 Graph Inequalities on a Number Line Guides

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March 21, 2022

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Kelli

About Kelli

Kelli Bhattacharjee is the owner of Freebie Finding Mom. When she’s not goofing around with her son, she’s busy blogging, or just hanging out with the family which usually involves listening to music too loud and having dance parties.

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Negative and Positive Number Line

Number lines

ByCristina Morero

This post may contain affiliate links, which means I’ll receive a commission if you purchase through my links, at no extra cost to you. Please read full disclosure for more information.

Looking for a printable negative and positive number line? Super! Here you can find multiple cute & free number line printables and blank worksheets in black & white – both landscape and portrait format. All PDFs are free and instantly downloadable.

Negative and positive number line worksheets found in this post

Number lines are a great way to understand numbers, learn how to count, and a fun way to practice basic addition and subtraction.

I’ve created multiple cute and simple number line printables you can print out for your children or students.

HOW TO PRINT: These printables are easy to print out and use – just follow these steps to get your worksheets printed:

Scroll down until you find the design you’d like to print out
Click on the instant download link under the image of the PDF
An ad might pop up, so close the ad, and the PDF will appear
Save the PDF to your computer and print them out.

Psst! The image has my logo on a banner, but no worries, the PDFs come without it.

Size: US Letter – but some can be easily resized and printed as A4.

Tip for printing: I always recommend saving the file to the computer first so that it will print out using the correct printer settings you have set up.

Here are the negative and positive number line printables. There are 18 options to choose from. All minimal and black & white to save some ink costs.

All the tables, charts, and math worksheets are copyright protected ©SaturdayGift Ltd. Graphics purchased or licensed from other sources are also subject to copyright protection. For classroom and personal use only, not to be hosted on any other website, sold, used for commercial purposes, or stored in an electronic retrieval system.

Positive and negative number line – portrait

Here you can find positive and negative number line printables in portrait form. The first one shows the decimals from -1 to 1, and the rest have whole numbers from -5 to 5 up to -1000 to 1000.

After the first printable the number lines show negative and positive integers. Any positive integer on the right side of the zero is on the positive number line, and any negative integers are on the negative number line.

Negative decimal number line from -1 to 1 – portrait

DOWNLOAD: Negative decimal number line from -1 to 1 – portrait

Number line with negatives to 5 – portrait

DOWNLOAD: Number line with negatives to 5 – portrait

Integer number line printable (to 10) – portrait

DOWNLOAD: Integer number line printable (to 10) – portrait

Positive and negative number line to 20 – portrait

DOWNLOAD: Positive and negative number line to 20 – portrait

Number line negative and positive numbers to 50 – portrait

DOWNLOAD: Number line negative and positive numbers to 50 – portrait

Negative and positive number lines up to 100 PDF – portrait

DOWNLOAD: Negative and positive number lines up to 100 PDF – portrait

Number line positive and negative from -500 to 500 – portrait

DOWNLOAD: Number line positive and negative from -500 to 500 – portrait

Number line negative and positive from 1000 to 1000 – portrait

DOWNLOAD: Number line negative and positive from 1000 to 1000 – portrait

Please note: I’ve added negative signs in front of the numbers showing negative numbers. The numbers with no sign indicate positive numbers (even though there’s no positive sign in front of them.)

Negative and positive number line printables – landscape

Here you can find positive and negative number line printables in landscape form.

Number line with negatives to 5 – landscape

Number line with negatives from -5 to 5 – landscape

DOWNLOAD: Number line with negatives from -5 to 5 – landscape

Integer number line printable (to 10) – landscape

Free printable number line (to 10) – landscape

DOWNLOAD: Free printable number line (to 10) – landscape

Number line negative and positive numbers to 50 – landscape

DOWNLOAD: Number line negative and positive numbers to 50 – landscape

Negative and positive number lines up to 100 PDF – landscape

DOWNLOAD: Negative and positive number lines up to 100 PDF – landscape

Number line positive and negative from -500 to 500 – landscape

DOWNLOAD: Number line positive and negative from -500 to 500 – landscape

Number line negative and positive from 1000 to 1000 – landscape

DOWNLOAD: Number line negative and positive from 1000 to 1000 – landscape

Only negative numbers have the negative sign in front of them. Positive numbers don’t have a positive sign to indicate they’re positive numbers. If needed, you can add the positive sign in front of the positive numbers.

Blank negative and positive numbers

Here you can find blank number lines. These are great resources for practicing skip-counting or doing basic addition or subtraction.

Blank negative and positive number lines landscape PDF

DOWNLOAD: Blank negative and positive number lines landscape PDF

Blank positive and negative numbers worksheet PDF

DOWNLOAD: Blank positive and negative numbers worksheet PDF

Blank printable number line negative and positive worksheet

DOWNLOAD: Blank printable number line negative and positive worksheet

Empty negative and positive number lines landscape PDF

DOWNLOAD: Empty negative and positive number lines landscape PDF

Adding positive and negative numbers worksheet PDF

Adding positive and negative numbers worksheet

DOWNLOAD: Adding positive and negative numbers worksheet

Blank number line – math worksheet printables

You can find more number line worksheets from this post.

COMING LATER – THE POST IS IN THE MAKING

Odd and even number – free printables

You can find free printable odd and even number sheets from this post. The colors help students identify numbers.

COMING LATER – THE POST IS IN THE MAKING

Number lines with negative numbers

You can find printable number lines with negative numbers and positive numbers from this post.

COMING LATER – THE POST IS IN THE MAKING

Number lines with fractions

You can find printable number lines with fractions from this post. Psst! You can also find the blank version for practicing.

COMING LATER – THE POST IS IN THE MAKING

Double number lines (blank)

You can find printable blank double number lines from this post. Double number lines are an excellent tool, for example, when given a price for a quantity of an item. And then, the student needs to know the price for more items.

CHECK OUT POST: Double number lines (free printable blank worksheets)

What is a negative and positive number line?

A number line is a visual aid and a simple and effective way to help children understand numbers and basic concepts of mathematics.

The positive and negative number line is a variation of the number line. The number line is a straight line with evenly spaced numbers.

The positive and negative number line has two lines, one for positive numbers and one for negative numbers. The zero or origin is in the center, and on the right side of the zero is the positive side and on the left side of the zero is the negative number line.

How do you use a negative and positive number line?

The negative and positive number lines can help children understand and visualize negative and positive numbers. The number line can also be helpful for addition and subtraction.

Examples:

Example 1: Adding positive numbers to negative numbers

-5 + 3

Go to the place on the number line where it says -5. Then move 3 steps towards the positive number line. The answer then is -2.

Example 2: Subtract the negative number from a positive whole number

+3 – 5

Go to the spot on the number line where it says 3. Then move towards the negative side of the number line 5 “steps” (-5). The answer you land on is -2.

Note that on the positive number line, 4 is more “positive” than 3, so 4 is a larger number.

But on the negative number line, -4 is “more negative” than -3, so -4 is smaller than -3.

Example 3: Subtract from a negative number

If you need to subtract from a negative, please note that the two negatives become positive, and the subtraction sign will cancel the negative sign.

-3 – -5 = -3 + 5

Example 4: Adding two negative numbers

-3 + -4

You’ll move further to the left on the number line when adding two negative numbers.

Other free printable math worksheets & resources

Looking for blank multiplication practice sheets? Check out this post: Multiplication charts 1-12 – cute & free printable grids.

Want to print out super cute times tables? Check out this page: All times tables from 0 times table to 12 times table.

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Cristina Morero

Cristina Morero. The creator of SaturdayGift, mindset coach, MBA, NLP Master Practitioner, planning and productivity expert, and a self-gifting extraordinaire.

The Number Line (Video & Practice)

TranscriptPractice

Hello, and welcome to this video about the number line! In this video, we will explore how to read number lines, draw and plot points on a number line, and add and subtract with a number line. Let’s get started!

We will use a number line to represent numbers in counting order. The number to the left is always smaller than the number to its right. To draw a number line, we first start by drawing a straight line and then we put arrows at the ends. The arrows represent the fact that the number line goes on forever in both directions. When dividing the number line with tick marks, it is important to place the tick marks equally spaced apart. This shows that the distance from any two numbers is the same. For example, the distance from 1 to 2 is the same as the distance from 2 to 3.

The most important number on a number line is 0 because every other number is based off of how far away it is from 0. Numbers to the left of 0 are negative, and numbers to the right of 0 are positive. Number lines continue infinitely in either direction, so the number line you see is always only part of the whole number line. Because of this, even though 0 is the most important number, it might not show up on your number line. For example, your number line might look like this if you are trying to plot the grade you got on your test.

Number lines can be used to represent a specific number. In the example we just mentioned, we can use the number line to represent a test grade of 94 by putting a dot above the number, like this:

Plotting a negative number would happen the same way. Simply put a dot over the number you are wanting to represent. The number line here shows the number -5.

We can also use a number line to help us add and subtract numbers. The movement to the right is for addition.

The movement to the left is for subtraction.

To use a number line for adding and subtracting, we start by plotting a point at our first number. Then we move right or left (depending on the operation) the number of spaces matching the second number.

Let’s practice by adding -7+5. We will start at -7, and since the operation is addition, we will move to the right 5 spaces. As you can see, we land on -2. Therefore, -7+5=-2.

Now let’s try subtracting 9-4. We will start at the number 9 on the number line. Since the operation is subtraction, this time we’re going to move to the left. We move left 4 spaces, which lands us on 5. Therefore, 9-4=5.

I hope that this video on number lines was helpful. Thanks for watching, and happy studying!

Question #1:

Which scenario matches the number line below?

Beth is selling bracelets at an art fair. She starts her day with nine bracelets, and then she sells four of them. Now she only has five more bracelets to sell.

Tony goes to the grocery store with nine dollars. He spends thirteen dollars for cereal and milk, and now he has four dollars left.

Tanya has thirteen dog treats in her pocket. She gives her puppy four of the treats, so now she has nine treats left.

John earns thirteen dollars mowing the neighbor’s lawn. On his way home, he spends nine dollars on a snack. Now John only has four dollars left.

Show Answer

Answer:

Tanya starts with thirteen treats in her pocket, and then gives her puppy four of them. This is represented on the number line as four jumps back to the left from the number thirteen, landing on the number nine. In other words, \(13-4=9\).

Hide Answer

Question #2:

Which math sentence matches the number line below? Note that the original starting point is at zero.

\(0+12-5=7\)

\(0-7+12=5\)

\(0+5-12=7\)

\(0-5+12=7\)

Show Answer

Answer:

Starting at 0, we move left (back) five jumps and land on -5. From -5, we then move to the right 12 jumps, landing on 7. Zero minus five, plus twelve, equals seven, which matches the number sentence \(0-5+12=7\).

Hide Answer

Question #3:

Which number line correctly plots the point 1.5?

Show Answer

Answer:

Number line B shows the point 1.5 correctly plotted between 1 and 2.

Hide Answer

Question #4:

Joey paints 11 picture frames blue, and 8 picture frames green. He paints a total of 19 picture frames. A number line is used to represent the total number of frames painted. Identify the error in the number line below.

The green line is incorrect. It shows a jump of 9, but it should show a jump of 8.

The blue line is incorrect. It should show a jump of 12, not 11.

Both lines are incorrect.

The green line is incorrect. It should show a jump of 13, not 9.

Show Answer

Answer:

Joey painted 11 blue frames, so the blue line is correct in showing a jump of 11. He painted 8 frames green, so the green line should show a jump of 8, from 11 to 19. However, the green line is incorrect. The green line shows a jump of 9, landing on 20.

Hide Answer

Question #5:

Fill in the missing value so that the number line matches the number sentence:
\(3+\text{___}=8\)

Show Answer

Answer:

The number line shows a starting point of 3, and then 5 jumps to the right, eventually landing on the number 8. The number line matches the number sentence \(3+5=8\), so the missing value is 5.

Hide Answer

Return to Algebra I Videos

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Algebra Topics: Negative Numbers

Lesson 3: Negative Numbers

/en/algebra-topics/exponents/content/

What are negative numbers?

A negative number is any number that is less than zero. For instance, -7 is a number that is seven less than 0.

-7

It might seem a little odd to say that a number is less than 0. After all, we often think of zero as meaning nothing. For instance, if you have 0 pieces of chocolate left in your candy bowl, you have no candy. There’s nothing left. It’s difficult to imagine having less than nothing in this case.

However, there are instances in real life where you use numbers that are less than zero. For example, have you ever been outside on a really cold winter day when the temperature was below zero? Any temperature below zero is a negative number. For instance, the temperature on this thermometer is -20, or twenty degrees below zero.

You can also use negative numbers for more abstract ideas. For instance, in finances negative numbers can be used to show debt. If I overdraw my account (take out more money than I actually have), my new bank balance will be a negative number. Not only will I have no money in the bank—I’ll actually have less than none because I owe the bank money.

Watch the video below to learn more about negative numbers.

Any number without a minus sign in front of it is considered to be a positive number, meaning a number that’s greater than zero. So while -7 is negative seven, 7 is positive seven, or simply seven.

Understanding negative numbers

As you might have noticed, you write negative numbers with the same symbol you use in subtraction: the minus sign ( — ). The minus sign doesn’t mean you should think of a number like -4 as subtract four. After all, how would you subtract this?

-4

You couldn’t—because there’s nothing to subtract it from. We can write -4 on its own precisely because it doesn’t mean subtract 4. It means the opposite of four.

Take a look at 4 and -4 on the number line:

You can think of a number line as having three parts: a positive direction, a negative direction, and zero. Everything to the right of zero is positive and everything to the left of zero is negative. We think of positive and negative numbers as being opposites because they are on opposite sides of the number line.

Another important thing to know about negative numbers is that they get smaller the farther they get from 0. On this number line, the farther left a number is, the smaller it is. So 1 is smaller than 3. -2 is smaller than 1, and -7 is smaller than -2.

Understanding absolute value

When we talk about the absolute value of a number, we are talking about that number’s distance from 0 on the number line. Remember how we said 4 and -4 were the same distance from 0? That means 4 and -4 have the same absolute value. We represent taking the absolute value of a number with two straight vertical lines | |. For example, |-3| = 3. This is read «the absolute value of negative three is three.»

Something important to remember: even though negative numbers get smaller as they get further from 0, their absolute value gets bigger. For example, -10 is smaller than -6. However, |-10| is bigger than |-6| because -10 has a greater distance from 0 than -6.

Calculating with negative numbers

Using negative numbers in arithmetic is fairly simple. There are just a few special rules to keep in mind.

Adding and subtracting negative numbers

When you’re adding and subtracting negative numbers, it helps to think about a number line, at least at first. Let’s take a look at this problem: 6 — 7. Even though 7 is larger than 6, you can subtract it in the exact same way as any other number, as long as you understand there are numbers smaller than 0.

6 — 7 = -1

While the number line makes it easy to picture this problem, there’s also a trick you could have used to solve it.

First, ignore the negative signs for a moment. Just find the difference between the two numbers. In this case, it means solving for 7 — 6, which is 1. Next, look at your original problem. Which number has the highest absolute value? In this case, it’s -7. Because -7 is a negative number, our answer will be one too: -1. Because the absolute value of -7 is greater than the distance between 6 and 0, our answer ends up being less than 0.

Adding negative numbers

How would you solve this problem?

6 + -7

Believe it or not, this is the exact same problem we just solved!

This is because the plus sign simply lets you know you’re combining two numbers. When you combine a negative number with a positive one, the sum will be less than the original number—so you might as well be subtracting. So 6 + -7 is the same thing as 6 — 7, and they both equal -1.

6 + -7 = -1

Whenever you see a positive and negative sign next to each other, you should read it as a negative. Just like 6 + -7 is the same as 6 — 7:

10 + -11 is equal to 10 — 11.
3 + -2 is equal to 3 — 2.
50 + -100 is equal to 50 -100.

This is true whenever you’re adding a negative number. Adding a negative number is always the same as subtracting that number’s absolute value.

Subtracting negative numbers

If adding a negative number is actually equal to subtracting, how do you subtract a negative number? For example, how do you solve this problem?

6 — — 3

If you guessed that you add them, you’re right. Here’s why: Remember how we said a negative number was the opposite of a positive one? We compared them to you and your mirror image. Your mirror image is your opposite, which means your mirror image’s opposite is you. In other words, the opposite of your opposite is you.

In the same way, you can simplify these two minus signs by reading them as two negatives. The first minus sign negates—or makes negative—the second. Because the negative—or opposite—of a negative is a positive, you can replace both minus signs with a plus sign. This means you’d solve for this:

6 + 3

This is a lot easier, to solve, right? If it seems confusing, you can just remember this simple trick: When you see two minus signs back to back, replace them with a plus sign.

So 6 minus negative 3 is equal to 6 plus 3. That’s equal to 9. In other words, 6 — -3 is 9.

Remembering all of the rules for adding and subtracting numbers can be overwhelming. Watch the video below for a trick to help you.

Multiplying and dividing negative numbers

There are two rules for multiplying and dividing numbers:

If you’re multiplying or dividing two numbers that are either both positive or both negative, your result will be positive.
If you’re multiplying or dividing a positive number and a negative number, your result will be negative.

That’s it! You multiply or divide as normal, then use these rules to determine whether the answer is positive or negative. For instance, take this problem, -3 ⋅ -4. 3 ⋅ 4 is 12. Because both numbers we multiplied were negative, the answer is positive: 12.

-3 ⋅ -4 = 12

On the other hand, if we were to multiply 3 ⋅ -4, we’d get a different answer:

3 ⋅ -4 = -12

Again, 3 ⋅ 4 is 12. But because one of our multiples is negative and the other is positive, our answer must also be negative: -12.

It works the same way for division. -40 / -10 is 4 because —40 and -10 are both negatives. However, -40 / 10 is -4 because one number is negative and the other is positive.

Continue

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Addition and subtraction of negative and positive numbers. Solving examples.

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Adding and subtracting negative numbers

There are different types of numbers — even numbers, odd numbers, prime numbers, composite numbers. Also, based on the sign of the number, there can be two types — positive numbers and negative numbers . These numbers can be represented on a number line. The average number in this row is zero. Negative numbers are on the left side of zero, and positive numbers are on the right side.

Zero is the neutral element with respect to integer addition. Basically in this article, we will study the operations of addition and subtraction with negative numbers. There are certain rules for signs when adding and subtracting:

In order to add two negative numbers, you must add two numbers and put a minus sign.

\ ((-2)+(-3)=-5 \)

9001

\((-8)+4=4-8=-4\)

\(9+(-4)=9-4=5\)

For every number except \(0\) there is an opposite element, when summed with it, zero is formed: you should know the rule: a minus times a minus gives a plus. That is, if there are two minuses side by side, the sum is a plus.

\((-7)-(-6)=(-7)+6=(-1)\)

If the first number is positive and the second is negative, subtract according to the same principle as adding: look , which number is greater in absolute value, we subtract the smaller number from the larger one and put the sign of the larger number.

\(7-9=-2\) since \(9>7\)

Also don’t forget minus times minus gives plus:

\(7-(-9)=7+9=16\)

Problem 1

\(-36+15\)

\((-17)+(-45)\)

\(-9+(-1)\)

Solution:

\(4+(-5)=4-5=-1\)
\(-36+15=-21\)
\((-17)+(-45)\) \(=-17-45=-62\)
\(-9+(-1)=-9-1=-10\)

Problem 2. Calculate:

\(3-(-6)\)
\(-16-35\)
\(-27-(-5)\)
\(-94-(-61)\)

Solution:

\(3-(-6)=3+6=9\)
\(-16-35=-51\)
\(-27-(-5)=-27+5=-22\)
\(-94-(-61)=-94+61=-33\)

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Row indexing | Python

Access to characters in strings is based on an indexing operation — after the string or the name of the variable referring to the string, the position numbers of the required characters are indicated in square brackets.

It should also be understood that this very access is based on an offset, i.e. character distance from the left or right edge of the string. This distance is measured in integers and essentially determines the position number of the characters in the string — their index. The catch lies in the phrase « is measured in integers «, which means that the index can be both positive and negative: positive indices are counted from the left edge, and negative ones are counted from the right. Moreover, the character count from the left edge starts with \ (0 \), and from the right it starts with \(-1\) (minus one).

In the simplest case, one arbitrary character can be extracted:

 >>> 'string'[0]
's'
>>>
>>> 'string'[5]
'g'
>>>
>>>
>>> s = 'STRING'
>>>
>>>s[0]
'S'
>>>
>>> s[5]
'G' 9'

And we can also extract characters from a string or slice with a specified step:

 >>> s = 'a0-b1-c2-d3-e4-f5-g6'
>>>
>>>s[::3]
'abcdefg'
>>>
>>>s[1::3]
'0123456'
>>>
>>>s[2::3]
'------'
>>>
>>>
>>> s[3:12:3]
'bcd'
>>> s[9:1:-3]
'dcb'

As you can see, on the one hand, everything seems to be simple, but on the other hand, especially looking at the latest examples, it becomes clear that there are still some difficulties. Let’s take it in order.

Character Extraction — S[i]

Extracting a single character from a string is indeed the simplest case. Let’s look at the line again:

 >>> s = 'STRING'

To extract the characters ‘ S ‘, ‘ R ‘ and ‘ G ‘ we can perform the following simple operations:

 >>> s[0], s[2], s[5]
('S', 'R', 'G')

But since we already know that the indices can be negative, i.e. count from the end of the string, then to extract the same characters, we can perform the following operations:

 >>> s[-6], s[-4], s[-1]
('S', 'R', 'G')

Negative and positive character indices can be expressed in terms of the length of the string:

 >>> s[4], s[-2]
('N', 'N')
>>>
>>> s[len(s) - 2], s[4 - len(s)]
('N', 'N')

Visually, the mapping of indices to characters of a string looks like this:

If you specify an index that is outside the string, it will result in an error:

 >>> s[10], s[-10]
Traceback (most recent call last):
  File "", line 1, in 
IndexError: string index out of range

The absence of an index is also considered an error.

Slicing — S[i:j]

Extracting string slices is done at two indexes, which are separated by a colon:

 >>> s = '*a*bb*ccc*dddd*'
>>>
>>> s[1:2]
'a'
>>>
>>> s[3:5]
'bb'
>>>
>>>s[6:9]
'ccc'
>>>
>>> s[10:14]
'dddd'

Note that the slice starts at the character whose index is listed first and ends before the character whose index is listed second:

The same results can be achieved by specifying negative indexes:

 >>> s[-14:-13], s[-12:-10], s[-9:-6], s[-5:-1]
('a', 'bb', 'ccc', 'dddd')

In all previous examples, the first index was always less than the second, but if it suddenly turns out to be the other way around, this will not lead to an error, but will return an empty string:

 >>> s[5:1]
''
>>> s[-10:-14]
''

And even if the indices are equal, we will still see an empty string instead of an error message:

 >>> s[1:1]
''
>>>s[-14:-14]
''

Will there be an error message if no indexes are specified at all? No, instead we will see the entire line:

 >>> s[:]
'*a*bb*ccc*dddd*'

This behavior is due to the fact that the slice start and end indexes have default values: start is always \(0\), end is always len(s) . So s[:] is equivalent to s[0:len(s)] . Having these preset values is quite handy if we want to slice from the beginning of a string to a specified character, or from some character to the end of a string:

 >>> s = '*a*bb*ccc*dddd*'
>>>
>>> s[:2] # equivalent to s[:-13]
'*a'
>>>
>>> s[:5] # equivalent to s[:-10]
'*a*bb'
>>>
>>> s[:-1] # equivalent to s[:14]
'*a*bb*ccc*dddd'
>>>
>>> s[-3:] # equivalent to s[12:]
'dd*'
>>>
>>> s[-10:] # equivalent to s[5:]
'*ccc*dddd*'

Again, note that if a slice is taken from the beginning of the string to the specified character, then the specified character itself is not included in the slice. And if the slice is taken from the specified character to the end of the string, then that character will be included at the beginning of the slice. This behavior guarantees that the concatenation of two such slices will be equal to the original string:

 >>> s = 'xxxxYYYY'
>>>
>>> s[:4] + s[4:]
'xxxxYYYY'
>>>
>>> s[:-3] + s[-3:]
'xxxxYYYY'

The slicing operation allows the specification of indices that go beyond the limits of the string:

 >>> s = '*a*bb*ccc*dddd*'
>>>
>>>s[:100]
'*a*bb*ccc*dddd*'
>>>
>>> s[6:100]
'cc*dddd*'
>>>
>>> s[100:]
''

Extracting slices with a given step — S[i:j:k]

To extract characters from a slice with a given step (they are also called sparse slices), you must specify the step value separated by a colon after the slice extraction operation:

 >>> s = 'A1-B2-C3-D4-E5-'
>>>
>>> s[3:12:3]
'BCD'
>>>
>>>s[:9:2]
'A-2C-'
>>>
>>>s[3::4]
'B3-'
>>>
>>>s[::6]
'ACE'

Obviously, we will achieve the same results if we do not change the step values, and change the values of the indexes of the beginning and end of the slice to negative ones:

 >>> s[-12:-3:3], s[:-6:2], s[-12::4]
('BCD', 'A-2C-', 'B3-')

Visually, all this can be represented like this:

The returned string contains only those characters of the original string that are marked in orange. The characters marked in green are part of the specified slice, but do not correspond to the stride value, so we do not see them in the resulting string.

The stride value has a default value of \(1\) — this corresponds to extracting all elements in a row. And given that the beginning of the slice and its end also have default values equal to \(0\) and len(s) , then we can do nothing at all (except for two colons, of course):

 >>> s = 'A1-B2-C3-D4-E5-'
>>>
>>>s[::]
'A1-B2-C3-D4-E5-'

Can the step value be negative? Yes:

 >>> 'ABCDEFG'[::-1]
'GFEDCBA'
>>>
>>> 'ABCDEFG'[::-2]
'GECA'

In fact, the sign of the specified number in the step indicates the direction in which the characters are extracted from the slice. The value \(-1\) means that each subsequent character should be extracted, moving from the right edge to the left. And the value \(-2\) is to extract every second character, also moving from the end to the beginning of the string.

Let’s run some examples and see how this can be visualized:

 >>> s = 'ABCDEFGHIJKLMNO'
>>>
>>>s[::2]
'ACEGIKMO'
>>>
>>>s[::-2]
'OMKIGECA'
>>>
>>>s[::4]
'AEIM'
>>>
>>>s[::-4]
'OKGC'

Remember, we said that the index of the start of a slice must be less than the index of its end? In fact, this need is related to the direction in which slice characters are extracted. By default, the stride value is \(1\) — this corresponds to the removal of each character in the direction from the beginning of the line to the end of it. This is why we previously got an empty string when we specified the left index greater than the right one:

 >>> s = 'ABCDEFGHIJKLMNO'
>>>
>>> s[10:4] # default stride is 1
''

Looking at the empty string returned by the interpreter and the picture, it becomes clear that something is wrong. We can clearly see that the extraction starts at index number \(10\), i.e. from the character » K » and even though the extraction is from left to right (because the default stride is \(1\)), the character » K » just has to get into the resulting string. But it’s not there!!!

It’s time to realize that character indexes in a string and their offsets are not the same thing. When working with sequences, it is easiest for us to perceive character indices (numbers of their positions) than their offsets. I’m even sure that after you figure out the offsets, you’ll still be imagining indexes.

Indexes and character offsets

Well, let’s make a small digression and try to figure it out. Here is the line we already know:

 >>> s = 'STRING'

And this is how we map indexes to characters:

Why are we doing this? Because it’s more convenient than offsets:

Speaking of offsets, we mean the left border of the symbols (the left side of the squares in which they are shown in the figures). So, for example, the character offset » S » relative to the left edge of the string is equal to \(0\) i.e. it is not shifted at all, so the operation s[0] returns it. And the offset of the character » G » relative to the right edge is \(- 1\), so the operation s[-1] returns » G «. All other indexing operations are based on the same principle, whether they are slices or slices with a given step.

Even if you understand what character offsets are, you will still rely on their indices. With your eyes, mouse pointer, finger, you will still be counting directly the characters themselves, and not their left borders. Just because it’s convenient. Simply because the vast majority of it works great and doesn’t cause any problems.

Now that we know what offsets are, it’s much easier for us to understand why the operation s[4:2:1] returns an empty string instead of the character » I «:

Looking at this picture, it starts to seem to us that the character » I » with the index \(4\) should get into the resulting string. But the Python interpreter is guided not by indices, but by offsets, and from its point of view, everything looks like this:

Since the symbol » I » is located before the offset with the number \(4\), and the count is based on offsets, it should not be in the resulting line either.

Understanding that strings are indexed by character offsets sheds light on many nuances. For example, taking a slice from some character to the end of the string:

 >>> s[2:6] # equivalent to s[2:]
'RING'

If we look at this operation in the context of indices, then we can argue that we are specifying an element at index \(6\), which does not actually exist, but before which the extraction of characters from the string should stop.

If you look at this operation in the context of displacements, then you don’t have to talk about some non-existent things. All of these offsets exist and are what the Python interpreter works with.

Most likely, you have a question about how the operation s[::-1] works, because its left index is \(0\) by default, and its right index is len(s) , but even though to the fact that the extraction of characters should be performed from right to left, i.e. we, in theory, should see an empty line, we see how everything works fine:

 >>> s = 'STRING'
>>>
>>>s[::-1]
'GNIRTS'
>>>
>>>s[::-2]
'GIT'

Such an exception to the rule was created intentionally, namely, for convenience. Because it’s really more convenient than writing something like s[len(s):0:-1] or s[len(s):0:-2] every time.

Negative step value

If we want to extract elements from string slices in reverse order, then the slice boundaries must also be specified in reverse order:

 >>> s = 'ABCDEFGHIJKLMNO'
>>>
>>> s[11:2:-2]
'LJHFD'
>>>
>>>s[:4:-3]
'OLIF'
>>>
>>> s[10::-4]
'KGC'
>>>
>>> s[-3:-6:-7]
'M'

Remember we wondered how the operation s[::-1] works? And a vaguely vague answer was given that this was supposedly just a technical feature created for convenience. So this is a really technically implemented trick, which consists in the fact that the default values \u200b\u200bof the start and end of the slice (start — \(0\), end — len(s) ) are swapped if the stride value is negative:

 >>> s[::2], s[0:len(s):2]
('ACEGIKMO', 'ACEGIKMO')
>>>
>>> s = 'ABCDEFGHIJKLMNO'
>>> s[::-2], s[-1:-len(s)-1:-2]
('OMKIGECA', 'OMKIGECA')

You may have been a little surprised when you expected s[len(s):0:-2] instead of s[-1:-len(s)-1:-2] . But if we remember that characters are extracted not by indexes, but by offsets, also remembering that slice extraction is performed from of the first specified offset to of the second, it will become clear that the command s[len(s):0:-2] will not give you what you need:

 >>> s[len(s):0:-2], s[::-2], s[-1:-len(s)-1:-2]
('OMKIGEC', 'OMKIGECA', 'OMKIGECA')

In conclusion, I want to remind you of two simple rules from the Zen of Python:

Simple is better than complex;
Complicated is better than complicated.

Why these rules. Because you may want to use some kind of cleverly invented tricks with indexing in your code. On the one hand: «Why not??? After all, programming is also a way of self-expression!». But on the other hand, it is still a wonderful way to take out the brain and yourself, and worst of all, other people. So, if suddenly, you come up with some kind of trick with row indexing, then do not be too lazy to reveal its secret in the comments to the code.

The indexing mechanism itself is quite simple. Nevertheless, even from such «simplicity» you can mold something like this:

 >>> sym = 'ABCDEFGHIJ'
>>> num = '1234567890'
>>> i = 2
>>>
>>> sym[int(num[i])-len(sym):int(num[len(sym)-i])-i:int(num[i])]
'DG'

Such tricks have every right to exist, but it is better to leave them for programming championships.

62.

Dual simplex method

The dual simplex method, like the simplex method, is used to find a solution to a linear programming problem written in the form of a main problem, for which among the vectors , composed of coefficients of unknowns in the system of equations, there is T single. At the same time, the dual simplex method can be used to solve a linear programming problem, the free members of the system of equations of which can be any numbers (when solving the problem by the simplex method, these numbers were assumed to be non-negative). We now consider such a problem, assuming previously that the vectors are unit, i.e., consider the problem of determining the maximum value of the function

(54)

Under the conditions

(55)

(56)

Where

And among the numbers there are negative ones.

In this case, there is a solution to the system of linear equations (55). However, this solution is not a plan for problem (54) — (56), since there are negative numbers among its components.

Since the vectors — are single, each of the vectors can be represented as a linear combination of these vectors, and the coefficients of the expansion of vectors into vectors are numbers. Thus, we can find

Definition 1.13. The solution of the system of linear equations (55), determined by the basis , is called Pseudoplan of problem (54) — (56), if for any

Theorem 1.13. If in the pseudoplan , Defined by the basis , there is at least one negative number such that all , Then the problem (54) — (56) has no plans at all.

Theorem 1.14. If there are negative numbers in the pseudo-plan , Defined by the basis , such that for any of them there are numbers , Then we can switch to a new pseudo-plan, in which the value of the objective function of the problem is (54) — (56 ) Will not decrease.

The formulated theorems provide a basis for constructing an algorithm for the dual simplex method.

So, let’s continue the consideration of problem (54) — (56). Let be the pseudoplan of this task. Based on the initial data, a simplex table is compiled (Table 15), in which some elements of the vector column are negative numbers. If there are no such numbers, then the simplex table contains the optimal plan for problem (54) — (56), since, by assumption, all . Therefore, in order to determine the optimal task plan, provided that it exists, one should make an ordered transition from one simplex tableau to another until negative elements are excluded from the vector column. In this case, all elements must remain non-negative all the time ( T +1)-th row, i.e. for any

Thus, after compiling the simplex table, it is checked whether there are negative numbers in the vector column. If there are none, then the optimal plan of the original problem is found. If they are available (which we assume), then choose the largest negative number in absolute value. In the case when there are several such numbers, take one of them: let this number be Bl. The choice of this number determines the vector excluded from the basis, i.e. in this case, the vector 9 is derived from the basis0488 Pl. To determine which vector should be entered into the basis, we find , where

Let this minimum value be taken at , then the vector Р R is entered into the basis. The number is a resolving element. The transition to a new simplex table follows the usual rules of the simplex method. The iterative process is continued until there are no more negative numbers in the column of the vector P 0 . At the same time, the optimal plan of the original problem, and hence the dual one, is found. If at some step it turns out that in I th row of the simplex table (Table 15) in the column of the vector P 0 is a negative number Bi , and among the other elements of this row there are no negative numbers, then the original problem has no solution.

Thus, finding a solution to problem (54) — (56) by the dual simplex method includes the following steps:

1. Find the problem pseudoplan.

2. Check this pseudoplan for optimality. If the pseudo-plan is optimal, then a solution to the problem is found. Otherwise, either the problem is unsolvable, or they switch to a new pseudo-plan.

3. The enabling row is selected by determining the largest absolute value of the negative number of the column of the vector P 0 and the enabling column by finding the smallest absolute value of the ratio of elements ( M +1) — and the row to the corresponding negative elements of the enabling row .

4. Find a new pseudo-plan and repeat all actions starting from step 2.

1.17. Find maximum value of function Conditional

Solution. Let’s write the original linear programming problem in the form of the main problem: Find the Maximum of the function under the conditions

(58)

(59)

Let us compose the dual problem for the last problem. This is the problem, as a result of which it is required to find the minimum value of the function

(60)

Under the conditions

(61)

(62)

Choosing the vectors and as the basis, we compile a simplex table (Table 16) for the original problem (57) — (59).

From this table we see that the plan of the dual problem (57) — (59) is . With this plan, since there are two negative numbers (-4 and -6) in the column of the vector Р 0 table 16, and there are no negative numbers in the 4th row, then, in accordance with the algorithm of the dual simplex method, we pass to a new simplex table. (In this case, this can be done, since there are 9 vectors in the rows0488 Р 4 and Р 5 are negative numbers. If they were absent, then the problem would be unsolvable.) The vector excluded from the basis is determined by the largest negative number in the vector column Р 0 . In this case, this number is -6. Therefore, we exclude the vector Р 5 from the basis. To determine which vector needs to be entered into the basis, we find where0488 P 2 . Let’s move on to the new SimpLEX table (Table 17).

This table shows that a new plan of the dual problem has been obtained. In this plan, the value of its linear form is Thus, using the algorithm of the dual simplex method, an ordered transition from one plan of the dual problem to another has been made.

Since the column of the vector Р 0 of table 17 contains a negative number -7, we will consider the elements of the 2nd row. Among these numbers there is one negative -3/2. If there were no such number, then the original problem would be unsolvable. In this case, we pass to a new simplex table (Table 18).

As can be seen from Table 18, the optimal plans for the original and dual problems are found. They are and . With these plans, the values of the linear forms of the original and dual problems are equal:

1.18. Find the maximum value of the function under the conditions

Solution. Multiplying the first and third equations of the system of constraints of the problem by -1, as a result, we arrive at the problem of finding the maximum value of the function under the conditions

Taking vectors Р 3, Р 4 and Р 5 as a basis, we compile a simplex table (Table 19).

There are no negative numbers in the 4th row of Table 19. Therefore, if there were no negative numbers in the column of the vector P 0, then the optimal plan would be written in table 19. Since there are negative numbers in the indicated column and the same numbers are contained in the corresponding rows, we pass to the new simplex table (table 20). To do this, we exclude the vector 9 from the basis0488 Р 5 and introduce the vector Р 1 into the basis. As a result, we get a pseudoplan

Since there are no negative numbers in the row of the vector P 3, the original problem has no solution.

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The int data type, integers in Python.

Regular integers in Python are of type int and are written as strings of decimal digits. Type 9 integers0232 int (positive and negative) have unlimited precision, can take arbitrarily large values. Type int are an immutable object, performing an operation on integers, you get a new numeric object.

Integers support the following operations :

arithmetic operations;
bitwise operations;
comparison operations.

Type int in Python is represented by class int() , it allows :

to convert a string to an integer of type int given the specified base (base 10 decimal, base 16 hexadecimal, base 8 octal, and base 2 binary).
convert real numbers of type float to type int (discards the fractional part).
convert octal, hexadecimal and binary integer literals to type int

Class int() cannot convert to type there is no unambiguous way to convert this type of numbers.

a string with a floating point number (real number)



 Examples of converting objects to type 
 int  : 
 # Converting a string with a record
# integer in decimal form to type int
>>> int(' -3 ', base=10)
#3
# When converting decimal literals,
# written in lines, the base can be omitted
>>> int(' +5 ')
# 5
>>> int('-15_125')
#-15125
# Convert float to `int`
>>> int(3.23)
#3
>>> int(1.)
# one
>>> int(3.14e-10)
#0
# Octal literals and strings with them - to type int
>>> int(0o177)
#127
>>> int(' 0o177 ', base=8)
#127
# Hexadecimal literals and strings with them - to type int
>>> int(0x9ff)
#2559
>>> int(' 0x9ff ', base=16)
#2559
# Binary literals and strings with them - to type int
>>> int(0b101010)
#42
>>> int('0b101010', base=2)
#42
 
 Starting with Python 3.6, single underscores are allowed between digits and after any number system specifier to make it easier to visually judge the magnitude of a number. 
 >>> 100_000, 0x_FF_FF, 0o7_777
 # (100000, 65535, 4095)
 
 Integers can also be written as hexadecimal (base 16), octal (base 8), and binary (base 2). 

 Hexadecimal literals begin with a combination of the characters 0x or 0X followed by hexadecimal digits (0-9 and A-F). Hexadecimal digits can be entered in either lower or upper case. 
 Octal literals begin with a combination of the characters 0o or 0O (zero followed by an uppercase or lowercase 'o') followed by octal digits (0-7). 
 Binary literals begin with a combination of characters 0b or 0B followed by binary digits (0 - 1) 

 All of these literals create integer objects, they are just alternative ways of writing values. You can use the built-in functions  hex()  ,  oct()  and  bin()  

 
  int.bit_length()   : 
 to convert an integer to a string with representation in any of the three number systems. Returns the number of bits required to represent an integer in binary, excluding sign and leading zeros: 
 >>> n = -37
>>>bin(n)
# '-0b100101'
>>> n.bit_length()
#6
 
 Equivalent: 
 def bit_length(self):
    # binary representation: bin(-37) => '- 0b100101'
    s = bin(self)
    # remove leading zeros and minus sign
    s = s. lstrip('-0b')
    #len('100101') => 6
    return len(s)
 
 
  int.bit_count()   : 
 Returns the number of ones in the binary representation of the absolute value of an integer. New in version 3.10. 
 Example: 
 >>> n = 19
>>>bin(n)
'0b10011'
>>> n.bit_count()
#3
>>> (-n).bit_count()
#3
 
 Equivalent to: 
 def bit_count(self):
    return bin(self).count("1")
 
 
  int.to_bytes(length, byteorder, *, signed=False)   : 
 Returns an array of bytes representing an integer. Parameters  length  ,  byteorder  are mandatory: 
 -  length  specifies the required number of bytes, 
 -  byteorder  determines in what order to return bytes and have values  'big'  - from high to low,  'little'  - from low to high. 
 -  signed  specifies the use of two's complement to represent an integer. If  signed=False  and a negative integer is given, then  OverflowError  is thrown. 
 >>> (1024).to_bytes(2, byteorder='big')
#b'\x04\x00'
>>> (1024).to_bytes(10, byteorder='big')
# b'\x00\x00\x00\x00\x00\x00\x00\x00\x04\x00'
>>> (-1024).to_bytes(10, byteorder='big', signed=True)
# b'\xff\xff\xff\xff\xff\xff\xff\xff\xfc\x00'
>>> x = 1000
>>> x.to_bytes((x.bit_length() + 7) // 8, byteorder='little')
# b'\xe8\x03'
 
 If the specified bytes are not enough to represent a number, an OverflowError exception will be raised. To find out the byte order that the platform uses, use  sys.byteorder  . 
 
  int.from_bytes(bytes, byteorder, *, signed=False)   : 
 Returns an integer that corresponds to the specified byte array. 
 Parameters  bytes  and  byteorder  are required. 
 -  bytes  must be a byte-like object (byte strings, byte arrays, array.array, etc.) 
 -  byteorder  determines in what order to return bytes and have values  'big'  - from high to low,  'little'  - from low to high. 
 -  signed  specifies the use of two's complement to represent an integer. If  signed=False  and a negative white number is given, then  OverflowError  is thrown. 
 >>> int.from_bytes(b'\x00\x7f', byteorder = 'big')
#127
>>> int.from_bytes(b'\x00\x7f', byteorder = 'little')
#32512
>>> int.from_bytes(b'\xff\x81', 'big', signed = True)
#-127
>>> int.from_bytes([1, 0], 'big') # you can specify an "array" of bytes
#256
>>> int.from_bytes([255, 255], 'big')
#65535
 
 Format strings. format method | Python 3 for Beginners and Dummies 
 Sometimes (rather often, to be exact) there are situations when you need to make a string by substituting some data received during the program execution (user input, data from files, etc.) into it. Data substitution can be done using string formatting. Formatting can be done with the % operator, or with the format method. 
 If the substitution requires only one argument, then the value is the argument itself: 
>>> 'Hello, {}!'. format('Vasya')
'Hello Vasya!' 
 And if there are several, then the values will be all arguments with substitution strings (regular or named): 
>>> '{0}, {1}, {2}'.format('a', 'b', 'c')
'a, b, c'
>>> '{}, {}, {}'.format('a', 'b', 'c')
'a, b, c'
>>> '{2}, {1}, {0}'.format('a', 'b', 'c')
'c, b, a'
>>> '{2}, {1}, {0}'.format(*'abc')
'c, b, a'
>>> '{0}{1}{0}'.format('abra', 'cad')
'abracadabra'
>>> 'Coordinates: {latitude}, {longitude}'.format(latitude='37.24N', longitude='-115.81W')
'Coordinates: 37.24N, -115.81W'
>>> coord = {'latitude': '37.24N', 'longitude': '-115.81W'}
>>> 'Coordinates: {latitude}, {longitude}'.format(**coord)
'Coordinates: 37.24N, -115.81W' 
 However, the format method does more. Here is its syntax: 
 replacement field ::= "{" [field name] ["!" conversion] [":" specification] "}"
field name ::= arg_name ("." attribute name | "[" index "]")*
transformation ::= "r" (internal representation) | "s" (human representation)
specification ::= see below 
 For example: 
>>> "Units destroyed: {players[0]}". format(players = [1, 2, 3])
'Units destroyed: 1'
>>> "Units destroyed: {players[0]!r}".format(players = ['1', '2', '3'])
"Units destroyed: '1'" 9"
sign ::= "+" | "-" | " "
width ::= integer
precision ::= integer
type ::= "b" | "c" | "d" | "e" | "E" | "f" | "F" | "g" | "G" |
                  "n" | "o" | "s" | "x" | "X" | "%" 
 Alignment is performed using a filler character. The following alignment options are available: 



  Flag  
  Value  


 '<' 
 placeholders will default to the left (right) alignment of the object. 9' 
 Center alignment. 



 The Option "Sign" is used only for numbers and can take the following values: 



  Flag  
  

 '+'+' . 


 '-' 
 '-' for negative, nothing for positive. 


 'Space' 
 '-' for negative, space for positive. 



 Field "Type" can take the following values: 



  Type  
  Value 

 


 'o' 
 Octal number. 


 'x' 
 Hexadecimal number (lowercase letters). 


 'X' 
 Hexadecimal number (uppercase letters). 


 'e' 
 Floating point number with exponent (lower case exponent). 


 'E' 
 Floating point number with exponent (upper case exponent). 


 'f', 'F' 
 Floating point number (normal format). 


 'g' 
 Floating point number. with the exponent (lower case exponent) if it's less than -4 or precision, otherwise normal format. 


 'G' 
 Floating point number. with exponent (exponent in upper case) if it is less than -4 or precision, otherwise normal format. 


 'c' 
 Character (string of one character or number - character code). 


 's' 
 string. 


 '%' 
 The number is multiplied by 100, a floating point number is displayed followed by a %. 



 And finally, a few examples: 930}'.format('centered') # use '*' as a fill char

'***********centered***********'

>>> '{:+f}; {:+f}'.format(3.14, -3.14) # show it always

'+3.140000; -3.140000'

>>>'{:f}; {: f}'.format(3.14, -3.14) # show a space for positive numbers

'3.140000; -3.140000'

>>> '{:-f}; {:-f}'.format(3.14, -3.14) # show only the minus -- same as '{:f}; {:f}'

'3.140000; -3.140000'

>>> # format also supports binary numbers

>>> "int: {0:d}; hex: {0:x}; oct: {0:o}; bin: {0:b}".format(42)

'int: 42; hex: 2a; oct: 52; bin: 101010'

>>> # with 0x, 0o, or 0b as prefix:

>>> "int: {0:d}; hex: {0:#x}; oct: {0:#o}; bin: {0:#b}".format(42)

'int: 42; hex: 0x2a; oct:0o52; bin: 0b101010'

>>> points = 19.5

>>> total = 22

>>> 'Correct answers: {:.2%}'.format(points/total)

'Correct answers: 88. 64%' 
  To insert Python code into a comment, wrap it in tags  

Your code
   
 5 Ways to check if a string is an integer in Python 
 Original by Team Python Pool. 
 Many times while doing some projects or maybe simple programming, we need to restrict whether a given Python string is an integer or not. So, in this detailed article, you will learn five dominant ways to check if a given python string is an integer or not. 
 So, without wasting time, let's jump straight into python's ways to check if a string is an integer. 
 Some Elite Python Ways to Check if a String Is Integer 

 isnumeric function 
 exception handling 
 isdigit function 
 Regular expression 
 any() and map() function 
 1.902 or Input string Integer or Does not use the isnumeric function 
 Python's isnumeric() function can be used to check if a string is an integer or not. isnumeric() is a built-in function. It returns True if all characters are numeric, False otherwise. 
 Syntax 
  string  .isnumeric() 
 Parameters 
 The isnumeric() method takes no parameters. 
 Examples 
 #1
print(s.isnumeric())
#2
print(s.isnumeric())
#3
print(s.isnumeric())
#four
if s. isnumeric():
   print('Integer')
else:
   print('Not an integer') 
 Exit 
 True
True
False
Integer 
 Explanation: 
 Explanation: 

 In the first example, we initialized and declared the string s with the value '69544'. After that, using the isnumeric() function, we checked if '69544' is an integer or not. In this case it is an integer so and it returned 'True'. 
 In the second python test example, if the string is an integer, we have initialized the string s with the value '\u00BD'. This '\u00BD' is a Unicode value and you can write digits and numeric characters using Unicode in the program. So it returns true. 
 The third example is similar to the first, but instead of declaring an integer value, we concatenated both the integer value and the string. In this case, the isnumeric() function will return False. 
 In the fourth example, we took a few extra steps using if-else with the isnumeric() function merged. Here we have declared and initialized our variable 's' with the value '5651'. Then, using flow control statements and the isnumeric() function, we checked whether the given string is an integer or not. In this case, it's an integer. Thus, we will get the output integer. In other cases, if the value is not an integer, then we will get a result saying "Not an integer". 

 Note: This method of checking if a string is an integer in Python will not work on negative numbers. 
 2. Python Checks Whether a String Is Integer Using Exception Handling 
 We can use python to check if a string is integer using the exception handling mechanism. If you don't know how an exception is handled in python, let me briefly explain it to you. In Python, exceptions can be handled using the try statement. A vital operation that can throw an exception is placed in a try clause. The code that manages exceptions is written in the except clause. This way we can choose which operations to do as soon as we catch the exception. 
 Let's see how this works with an example. 
 Syntax 
 try:
    #Code
except:
    # Code 
 Parameters 
 The exception handling mechanism (try-except-finally) takes no parameters. 
 Examples 
 Exit 
 Not an integer 
 Explanation: 
 In the above example, we initialized the string 's' with the value '951 sd'. Initially, we assume that the value of string 's' is an integer. So we declared it to be true. After that, we tried to convert the string to an integer using the int function. If the string's' contains non-numeric characters, then ' int' will throw a ValueError indicating that the string is not an integer, and vice versa. 
 Also, along with the exception handling mechanism, we used flow control statements to print the output appropriately. 
 Note: This method of checking if a string is an integer in Python will also work with Negative numbers. 
 3. Python Checks Whether a String Is an Integer Using the isdigit Function 
 We can use the isdigit() function to check if a string is an integer or not in Python. The isdigit() method returns True if all characters in the string are digits. Otherwise it returns False. 
 Let's see how this works with an example. 
 Syntax 
 string.isdigit() 
 Parameters 
 The isdigit() method takes no parameters. 
 Return value of the isdigit() function 

 Returns True - If all characters in the string are digits. 
 Returns False - If the string contains one or more non-numeric digits 

 Examples

User input is Integer 
 Explanation: 
 A third example of checking whether an input string is an integer is using the isdigit() function. Here, in the example above, we have taken the input from a string and stored it in the variable 'str.' After that, using the control statements and the isdigit() function, we checked whether the input string is an integer or not. 
 Note: The 'isdigit()' function will only work for positive integers, i.e. if you pass any floating point number, it will say it's a string. It doesn't take any arguments, so it returns an error when passed as parameter 
 4. Python Checks Whether a String Is an Integer Using Regular Expression 
 We can use a search pattern which is known as a regular expression to check if a string is an integer or not in Python. If you don't know what a regular expression is and how it works in python, let me briefly explain it to you. In Python, a regular expression is a specific sequence of characters that allows you to match or find other strings or sets of strings with a specialized syntax held in a pattern. Regular expressions are widely used in the UNIX world. 
 Here we use the match method of the regular expression, that is, re.match().Re. match() only searches the first line of the string and returns a match object if found, otherwise returns none. But if the substring match is in some other string than the first string of the string (in the case of a multiline string), it returns none. 
 Let's see how it works with an example. 
 Syntax 
 re. match(pattern, string,) 
 Parameters 

 pattern Contains the regular expression to be matched. 
 string It consists of a string that will be searched according to the pattern at the beginning of the string. 
 flags (optional) You can specify different flags with a bitwise OR (|). These are modifiers. 

 Return value 

 Return matching objects if found. 
 If there is no match, None will be returned instead of a match href="https://en.wikipedia.org/wiki/Object_(computer_science)">Object. href="https://en.wikipedia.org/wiki/Object_(computer_science)">Object. 

 Examples 
 import re("Enter any value: ")
. match("[-+]?\d+$", value)
if result is not None:
    print("User input is an Integer")
else:
    print("User Input is not an integer") 
 Exit 
 Enter any value: 965oop
User Input is not an integer 
 Explanation: 
 The fourth way to check if an input string is an integer or not in Python is to use the regular expression engine. In this example, we first imported the regular expression with 'import re'. After that, we took the input from the user and stored it in a variable value. We then used our re.match() method to check if the input string is an integer or not. The pattern to be matched here is "[ - +]?\d+$". This pattern specifies that it will only match if we have an input string as an integer. 
 Note: The 're.match()' function will also work with negative numbers. 
 5. Python Checks Whether a String Is an Integer Using the any() and map() Functions 
 We can use a combination of the any() and map() functions to check if a string is an integer or not in Python. If you don't know what any() and map() functions are and how they work in python, let me briefly explain it to you. 

 The any() function accepts an iteration (list, string, dictionary, etc.) in Python. This function returns true if any element in the iterable is true, otherwise it returns false. 
 The map() function calls the specified function for each element of an iterable (for example, a string, list, tuple, or dictionary) and returns a list of results. 

 Let's look at examples of how they work. 
 Syntax 
 Syntax of any function() 
 any(iterable) 
 Syntax of the map() function 
 map(function, iterable [ iterable2, iterable3,. ..iterableN]) 
 Function parameters any 

 () 
 iterable: Iterable object (list, tuple, dictionary) 
 Map() function parameters 
 Function: A function to be executed for each element of an iterable Sequence, collection, or iterator object. You can submit as many iterations as you like, just make sure the function has one parameter for each iteration. 
 Return Value 

 Any: The any() function returns True if any element in the iterable is true, otherwise it returns False. 
 Map: Returns a list of results after applying this function 

 Examples 
 Output 
 Explanation: 
 A fifth way to check if an input string is an integer or not in Python is to use a combination of the any() and map( functions ) in python. Here in the example above, we have taken the input as a string which is 'sdsd'. And after that, using the any(), map(), and isdigit() functions, we have to check if the string is an integer. 
 We get False because the input string is 'sd'. 
 Note: This method will also work with negative numbers. 
 Python applications Check if a string is an integer 

 To check if a given string variable or value contains only integers, for example, to check if a user entered a numeric parameter correctly in a menu-driven application. 
 Using ascii character values, count and print all digits using the isdigit() function. 

 Must Read 

 Introduction to Python Super With Examples 
 Python Help Function 
 Why is Python's sys.exit better than other exit functions? 
 Python Bitstring: Classes and other examples | Module 

 Output: Python Checks if a string is an integer 
 So if you follow this to the end, I'm pretty sure you can now understand all the possible ways to check if a string is an integer in Python. The best way to check if a string is an integer in Python depends on your needs and the type of project you're doing.



Miscellaneous

Flag	Value
'<'	placeholders will default to the left (right) alignment of the object. 9'	Center alignment.

Flag		'+'+' .
'-'	'-' for negative, nothing for positive.
'Space'	'-' for negative, space for positive.

Type	Value

'o'	Octal number.
'x'	Hexadecimal number (lowercase letters).
'X'	Hexadecimal number (uppercase letters).
'e'	Floating point number with exponent (lower case exponent).
'E'	Floating point number with exponent (upper case exponent).
'f', 'F'	Floating point number (normal format).
'g'	Floating point number. with the exponent (lower case exponent) if it's less than -4 or precision, otherwise normal format.
'G'	Floating point number. with exponent (exponent in upper case) if it is less than -4 or precision, otherwise normal format.
'c'	Character (string of one character or number - character code).
's'	string.
'%'	The number is multiplied by 100, a floating point number is displayed followed by a %.

example

try it

example

try it

Basic Rules for Positive and Negative Numbers

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Addition: Same Signs, Add the Numbers

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Understanding Math Foundations

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