Add positive and negative numbers: Adding Positive and Negative Numbers

Adding and Subtracting Positive and Negative Numbers

Numbers Can be Positive or Negative

This is the Number Line:

Negative Numbers (−)	Positive Numbers (+)

«−» is the negative sign.	«+» is the positive sign

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Example: 5 is really +5

Play with it!

On the Number Line positive goes to the right and negative to the left.

Try the sliders below and see what happens:

numbers/images/number-line-add.js?sub=n

Balloons and Weights

Let us think about numbers as balloons (positive) and weights (negative):

This basket has balloons and weights tied to it:

• The balloons pull up (positive)
• And the weights drag down (negative)

Adding a Positive Number

Adding positive numbers is just simple addition.

 

We can add balloons (we are adding positive value)

the basket gets pulled upwards (positive)

Example: 2 + 3 = 5

is really saying

«Positive 2 plus Positive 3 equals Positive 5»

We could write it as (+2) + (+3) = (+5)

Subtracting A Positive Number

Subtracting positive numbers is just simple subtraction.

We can take away balloons (we are subtracting positive value)

the basket gets pulled downwards (negative)

Example: 6 − 3 = 3

is really saying

«Positive 6 minus Positive 3 equals Positive 3»

We could write it as (+6) − (+3) = (+3)

Adding A Negative Number

Now let’s see what adding and subtracting negative numbers looks like:

We can add weights (we are adding negative values)

the basket gets pulled downwards (negative)

Example: 6 + (−3) = 3

is really saying

«Positive 6 plus Negative 3 equals Positive 3»

 

We could write it as (+6) + (−3) = (+3)

The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.

So these have the same result:

• (+6) − (+3) = (+3)
• (+6) + (−3) = (+3)

In other words subtracting a positive is the same as adding a negative.

Subtracting A Negative Number

Lastly, we can take away weights (we are subtracting negative values)

the basket gets pulled upwards (positive)

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Yes indeed! Subtracting a Negative is the same as adding!

Two Negatives make a Positive

What Did We Find?

Adding a positive number is simple addition …

Adding a Positive is Addition

 

Positive and Negative Together …

Subtracting a Positive
or
Adding a Negative
is
Subtraction

Example: What is 6 − (+3) ?

6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7) ?

5+(−7) = 5 − 7 = −2

 

Subtracting a negative .

..

Subtracting a Negative is the same as Adding

Example: What is 14 − (−4) ?

14−(−4) = 14 + 4 = 18

The Rules:

It can all be put into two rules:

	Rule				Example
+(+)	Two like signs become a positive sign				3+(+2) = 3 + 2 = 5
−(−)					6−(−3) = 6 + 3 = 9
+(−)	Two unlike signs become a negative sign				7+(−2) = 7 − 2 = 5
−(+)					8−(+2) = 8 − 2 = 6

They are «like signs» when they are like each other (in other words: the same).

 

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign

Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign.

5+(−2) = 5 − 2 = 3

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Starting Negative

What if we start with a negative number?

Using The Number Line can help:

Example: What is −3+(+2) ?

+(+) are like signs, so they become a positive sign.

−3+(+2) = −3 + 2

Start at −3 on the number line,
move forward 2 and you end up at −1

−3+(+2) = −3 + 2 = −1

Example: What is −3+(−2) ?

+(−) are unlike signs, so they become a negative sign.

−3+(−2) = −3 − 2

Start at −3 on the number line,
move back 2 and you end up at −5

−3+(−2) = −3 − 2 = −5

Now Play With It!

Try playing Casey Runner, you need to know the rules of positive and negative to succeed!

A Common Sense Explanation

And there is a «common sense» explanation:

If I say «Eat!» I am encouraging you to eat (positive)

If I say «Do not eat!» I am saying the opposite (negative).

Now if I say «Do NOT not eat!», I am saying I don’t
want you to starve, so I am back to saying «Eat!» (positive).

So, two negatives make a positive, and if that satisfies you, then
you are done!

 

Another Common Sense Explanation

A friend is +, an enemy is −

+ + ⇒ +		a friend of a friend is my friend
+ − ⇒ −		a friend of an enemy is my enemy
− + ⇒ −		an enemy of a friend is my enemy
− − ⇒ +		an enemy of an enemy is my friend

A Bank Example

Example: Last year the bank subtracted $10 from your account by mistake, and they want to fix it.

So the bank must take away a negative $10.

Let’s say your current balance is $80, so you will then have:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

A Long Example You Might Like

Ally’s Points

Ally can be naughty or nice. So Ally’s parents have said

«If you are nice we will add 3 points (+3).
If you are naughty, we take away 3 points (−3).
When you reach 30 Points you get a toy.»

 

Ally starts the day with 9 Points:		9
Ally’s Mom discovers spilt milk:		9 − 3 = 6
Then Dad confesses he spilt the milk and writes «undo». How do we «undo» a minus 3? We add 3 back again!
So Mom calculates:		6 − (−3) = 6 + 3 = 9

So when we subtract a negative, we gain points
(i. e. the same as adding points).

So Subtracting a Negative is the same as Adding

 

A few days later. Ally has 12 points.
Mom adds 3 points because Ally’s room is clean.		12 + 3 = 15
Dad says «I cleaned that room» and writes «undo» on the chart. Mom calculates:		15 − (+3) = 12
Dad sees Ally brushing the dog. Writes «+3» on the chart. Mom calculates:		12 + (+3) = 15
Ally throws a stone against the window. Dad writes «−3» on the chart. Mom calculates:		15 + (−3) = 12

See: both «15 − (+3)» and «15 + (−3)» result in 12.

So:

It doesn’t matter if you subtract positive points
or add negative points,
you still end up losing points.

So Subtracting a Positive
or
Adding a Negative
is
Subtraction

 

Try These Exercises …

Now try This Worksheet, and see how you go.

And also try these questions:

11715, 11716, 11717, 11718, 11719, 11720, 11721, 3445, 3446

Basic Rules for Positive and Negative Numbers

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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.

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.

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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.

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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.

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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).

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.

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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.

what they call it that, what is it in mathematics, what is the difference between

Content:

• What numbers are called positive and negative
• Comparing positive and negative numbers
• Rules for actions with negative and positive numbers
• Examples of problems with a solution

Contents

• What numbers are called positive and negative
• Comparing positive and negative numbers
• Rules for actions with negative and positive numbers
• Examples of problems with a solution

What numbers are called positive and negative

Negative numbers in algebra are numbers with a minus sign (-). For example, such numbers include -1, -2, -3. You can read the entry as minus one, minus two, minus three.

A negative number is any number less than zero preceded by a minus sign.

Positive numbers — numbers consisting of the set of positive numbers are numbers without a minus sign in the designation and are not zero.

Caution! If the teacher detects plagiarism in the work, major problems cannot be avoided (up to expulsion). If it is not possible to write yourself, order here.

In the system of negative numbers, as well as among the positive ones, there are fractions: ordinary and decimal, integers, roots, and so on. Almost all subtypes of numbers that are found among positive numbers are also found among negative ones. It is worth noting that, according to the concept, the number 0 is neither positive nor negative.

Positive numbers are numbers corresponding to points in that part of the coordinate line that lies on the right side of the origin.

Negative numbers — are numbers corresponding to points in the part of the coordinate line, which is located on the left side of the origin (zero).

A good example of the use of negative numbers is a thermometer. The device shows the temperature of the body, air, soil, water. In winter, when the weather is cold, the air temperature drops to negative values. For example, -10 degrees below zero:

Ordinary numbers, including 1, 2, 3 are called positive. These numbers have a (+) sign. Usually, it is not recorded.

Coordinate line — is a straight line on which all numbers are placed, including negative and positive.

The coordinate line has the following form:

In this case, only the numbers from −5 to 5 are marked. In fact, the coordinate line is infinite. On the image you can see only a fragment of this line. In order to mark numbers on the coordinate line, I use points. The reference point is zero. Negative numbers are marked to the left of zero, and positive numbers to the right.

The coordinate line continues indefinitely on both sides. Infinity in mathematics is represented by \(\infty\). The negative direction will be denoted by the symbol −\(\infty\), and the positive direction by the symbol +\(\infty\). Thus, the coordinate line contains all numbers from minus infinity to plus infinity:

\((−\infty; +\infty)\)

Each point on the coordinate line has a specific name and coordinate. The name is any Latin letter. A coordinate is a number indicating the position of a point on a line. Thus, the coordinate is the number that you want to mark on the coordinate line. For example, point A (2) is read as “point A with coordinate 2” and is indicated on the coordinate line as follows:

When considering the image of the coordinate line, you can see that negative numbers lie to the left of the origin, and positive numbers to the right. With each step to the left, the number will decrease downwards. With each step in the right direction, the number will increase.

Comparison of positive and negative numbers

Positive numbers, that is, those that are greater than 0, can be considered as a profit, an increase, an increase in the amount of something. Negative numbers can be represented as lack, loss, expense, debt. Suppose there are 55 items, such as apples. The number 55 is positive. In the case when you want to give someone 5 apples, this action can be denoted as -5. On a thermometer, an increase in temperature by 4.5 values ​​can be described as +4.5, and a decrease, in turn, as -4.5. In instruments that are used for measurements, positive and negative numbers are often used. This is due to the convenience of displaying changes in values.

Any negative number is less than any positive number. For example, if you compare -5 and 3, then minus five is less than three. This is because -5 is a negative number and 3 is a positive number. Using the coordinate line, it is quite easy to determine the position of these numbers.

On the straight line -5 is located to the left of the number 3. According to the rule, any negative number is less than any positive number. It follows that:

−5 < 3

Of two negative numbers, the one located to the left on the coordinate line is less. For example, when comparing the numbers -4 and -1, we can conclude that minus four is less than minus one. The reason is that on the coordinate line -4 is located more to the left than -1.

It can be seen that -4 lies to the left, and -1 to the right. Of the two negative numbers, the smaller one is the one located to the left on the coordinate line. Thus:

-4 < -1

Zero is greater than any negative number. For example, when comparing 0 and -3, you can conclude that zero is greater than minus three. This is due to the fact that on the coordinate line 0 is located to the right than -3.

When considering the coordinate line, you can see that 0 lies to the right, and -3 to the left. According to the rule, zero is greater than any negative number. Thus:

0 > -3

Zero is less than any positive number. For example, you can compare 0 and 4. Zero is less than 4.

On the coordinate line, 0 is to the left and 4 to the right. Based on the rule, zero is less than any positive number. Thus:

0 < 4

Rules for dealing with negative and positive numbers

The following sign rules exist for multiplying and dividing negative numbers:

1. .
2. When a positive number is multiplied or divided by a negative number, the result is a negative number.
3. If you want to multiply or divide a negative number by a positive number, you get a negative number.

In the process of adding negative numbers, one should be guided by similar rules of signs in a slightly different form. According to the general wording, the rule of signs sounds like this: “A plus on a minus gives a minus, a minus on a minus gives a plus, and a plus on a plus gives a plus. ” In this case, when adding a negative number to another, you get:

-а+(-в)=-а-в — that is, a positive number is subtracted from a negative number.

A similar rule applies to examples with the subtraction of negative numbers:

-а-(-в)=-а+в — a positive number is added to a negative number.

When you want to add two negative numbers, add the two numbers and put a minus sign. For example:

(−2)+(−3)=−5(−2)+(−3)=−5

If the first number is positive and the second is negative, you need to determine which number is greater in absolute value. Next, you need to subtract the smaller number from the larger one and put the sign of the larger number. For example:

(−8)+4=4−8=−4

9+(−4)=9−4=5

Every number except 0 corresponds to the opposite element. In sum, the number gives 0. For example:

−9+9=0

7.1+(−7.1)=0

When subtracting two negative numbers, you should be guided by the rule: minus times minus gives plus. Thus, when there are two minuses side by side, the sum is a plus. For example:

(−7)−(−6)=(−7)+6=(−1)

which is addition. It is necessary to determine which number modulo is greater. Next, subtract the smaller number from the larger one and put the sign of the larger number.

7−9=−2

since 9>7

solution

Task 1

Task

Need to solve: (+3) + (+4)

Solution:

(+3) + (+4) = +7

Answer: 7

Problem 2

Problem

Need to solve: (-4) + (-3)

Solution:

(-4) + (-3) = -7

Response: -7

Task 3

Task

Addition required: (+15) + (-7)

Solution:

(+15) + (-7) = 15 — 7 = 8

Answer: 8

4

Problem

Subtract: (+7) — (+4)

Solution:

(+7) — (+4) = +3

Answer: 3

Problem 5

Problem

Required to find the difference between numbers: -17 — (-14)

Solution:

-17 — (-14) = -17 + 14 = -3

Answer: -3

Problem 6

Problem

It is necessary to solve the example: (+5) ⋅ (-8)

Solution:

(+5) ⋅ (-8) = -40

Answer: -40

Problem 7

Problem

Find the product of two numbers: -9 ⋅ (-9)

Solution:

-9 ⋅ (-9) = 81

Answer: 81

Problem 8

Problem

Need to solve example: -6 ⋅ 5

Solution:

-6 ⋅ 5 = -30

Answer: -30

Problem 9

Problem

Divide two numbers: 40 : (-8)

Solution:

40 : (-8) = -5

Answer: -5

Problem 10

Problem

Required to find the difference: (-6) — (+6) — (-8)

Solution:

(-6) — (+6) — (-8) = -12 — (-8) = — 12 + 8 = -4

Response: -4

Task 11

Task

Need to solve example: (-5) ⋅ (-4) + (+3) ⋅ (-2)

Solution:

(-5) ⋅ (-4) + (+3) ⋅ (-2) = 20 + (-6) = 14

Answer: 14

Problem 12

Problem

Need to find the answer: (-15) ⋅ [-3 + (-15)] : (+5)

Solution:

(-15) ⋅ [-3 + (-15)] : (+5) = -15 ⋅ (-18) : 5 = (-15 : 5) ⋅ (-18) = -3 ⋅ (-18) = 54

Answer: 54

Task 13

Task

Division required: -18 : [-20 — (30 — 56)]

Solution:

-18 : [-20 — (30 — 56)] = -18 : [-20 — (-26) ] = -18 : (-20 + 26) = -18 : 6 = -3

Answer: -3

Problem 14

Problem

Need to find the value of the expression:

(−1)−(−512)⋅(+411)=(−1)−(−521)⋅(+114)

Solution:

(−1)−( −512)⋅411=−1−(−112)⋅411=(−1)−(−521)⋅114=−1−(−211)⋅114=−1−(−2)=−1+2 =1−1−(−2)=−1+2=1

Answer: 1

Task 15

Task

To be calculated:

Calculate |a| — |b| + |c|

with a = -8, b = -5, c = 1

Solution:

|-8| — |-5| + |1| = 8 — 5 + 1 = 4

Answer: 4

Problem 16

Problem

Need to solve an example:

[2. 4−(0.3−0.21)⋅2+0.44:(−2)]:45=[2.4−(0.3−0.21 )⋅2+0.44:(−2)]:54

Solution:

[2.4−(0.3−0.21)⋅2+0.44:(−2)]:45= [2.4−(0.3−0.21)⋅2+0.44:(−2)]:54=

[2.4−0.09⋅2+(−0.22)]:45=[2.4−0.09⋅2+(−0.22)]:54= (2.4−0 ,18−0.22):25=

2:45=52=2.5(2.4−0.18−0.22):52=2:54=25=2.5

Answer: 2.5

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Representation of positive and negative numbers in computer memory. Direct and additional code number

Direct code

The direct code is a representation of a number in the binary number system, in which the first (highest) digit is assigned to the sign of the number. If the number is positive, then 0 is written to the left bit; if the number is negative, then 1 is written to the left bit.

Thus, in the binary number system, using a direct code, a seven-digit number can be written in an eight-bit cell (byte). For example:

0 0001101 — positive number
1 0001101 — negative number

The number of values ​​that can be placed in a seven-digit cell with a sign in the extra bit is 256. This is the same as the number of values ​​that can be placed in an eight-digit cell without specifying a sign. However, the range of values ​​is already different, it includes values ​​from -128 to 127 inclusive (when converted to decimal notation).

At the same time, in computer technology, direct code is used almost exclusively to represent positive numbers.

For negative numbers, the so-called additional code is used. This is due to the convenience of performing operations on numbers by electronic devices of a computer.

Additional code

In the two’s complement code, as well as the direct one, the first digit is assigned to represent the sign of the number. The direct code is used to represent positive numbers, and the complementary code is used to represent negative numbers. Therefore, if there is a 1 in the first digit, then we are dealing with an additional code and with a negative number.

All other digits of the number in the additional code are first inverted, i.e. are reversed (0 to 1 and 1 to 0). For example, if 1 0001100 is a direct code of a number, then when forming its additional code, you must first replace zeros with ones, and ones with zeros, except for the first digit. We get 1 1110011. But this is not the final form of the additional number code.

Next, add one to the number obtained by inversion:

1 1110011 + 1 = 1 1110100

As a result, a number is obtained, which is usually called an additional number code.

The reason for using two’s complement to represent negative numbers is because it’s easier to do math. For example, we have two numbers represented in direct code. One number is positive, the other is negative, and these numbers must be added. However, you can’t just put them together. First, the computer must determine what these numbers are. Having found out that one number is negative, he should replace the operation of addition with the operation of subtraction. Then, the machine must determine which number is greater in modulo, in order to figure out the sign of the result and decide what to subtract from what. The result is a complex algorithm. It’s much easier to add numbers if the negative ones are converted to two’s complement. This can be seen in the examples below.

The operation of adding a positive number and a negative number represented in a direct code

1. Direct code number 5: 0 000 0101
   Direct code number -7: 1 000 0111
2. The two original numbers are compared. The sign of the larger original number is written to the sign bit of the result.
3. If the numbers have different signs, then instead of the operation of addition, the operation of subtraction from the larger value of the smaller value is used.