Does a negative times a negative equal a positive: Multiplying Positive and Negative Numbers: 3 Simple Rules

Negative Times a Negative

Minus times minus results in a plus, The reason for this, we needn’t discuss.                      — Ogden Nash The swimming pool Here is an anology of filling or draining a swimming pool: If you are filling the pool at a rate of 3 litres per second (+3) in 4 seconds (+4) the amount of water will have increased by 12 litres (+3 * +4 = +12). If you are draining the pool at a rate of 3 litres per second (-3) in 4 seconds (+4) the amount of water will have decreased by 12 litres (-3 * +4 = -12). If you are filling the pool at a rate of 3 litres per second (+3) 4 seconds ago (-4) the amount of water was 12 litres less (+3 * -4 = -12). If you are draining the pool at a rate of 3 litres per second (-3) 4 seconds ago (-4) the amount of water was 12 litres more than it is now (-3 * -4 = +12). This anology may provide students with an intuitive basis for the multiplication rules of positive and negative numbers. This example may not work for you, and you might want to read others below. Number Line Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y. Patterns Let’s see how a mathematician might understand what’s going on when a negative number is multiplied by a negative number. Here’s a plausibility argument drawn from multiplication patterns: 3 x -3 = -9 2 x -3 = -6 1 x -3 = -3 0 x -3 = 0 -1 x -3 = ? A Mathematical Explanation If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way: (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab For example, -2 * -3 = (-1)(2)(-1)(3) = (-1)(-1)(2)(3) = (-1)(-1) * 6 So the real question is, (-1)(-1) = ? and the answer is that the following convention has been adopted: (-1)(-1) = +1 This convention has been adopted for the simple reason that any other convention would cause something to break. For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn’t work for negative numbers: (-1)(1 + -1) = (-1)(1) + (-1)(-1) (-1)(0) = -1 + -1 0 = -2 As Sherlock Holmes observed, «When you have excluded the impossible, whatever remains, however improbable, must be the truth.» Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1. More Mathematical Explanation It may seems like someone just made the rule up out of thin air, with no particular reason why the answer should be positive. But there is a chain of reasoning — a mathematical «argument» — that shows why the rule *has* to be that negative times negative equals positive. Here’s how the reasoning goes: (1) Zero times anything equals zero. (2) Every number has exactly one additive inverse. This means if N is a positive number, then -N is its additive inverse, so that N + (-N) = 0. Likewise, the additive inverse of -N is N. (3) We want negative numbers to obey the distributive law. This says that a*(b+c) = a*b + a*c (4) Now, we are forced to accept a new law, that negative times positive equals negative. This is because we can use the distributive law on an expression like 2*(3 + (-3)). This equals 2*(0), which is zero. But by the distributive law, it also equals 2*3 + 2*(-3). So 2*(-3) does the job of the additive inverse of 2*3, and therefore 2*(-3) is the additive inverse of 2*3. But the additive inverse of 6 is just -6. So 2 times -3 equals -6. (5) Next, we are forced to accept another new law, that negative times negative equals positive. It’s a lot like the example in (4). We use the distributive law on, say, -3*(5 + (-5)). This is again equal to zero. But by the distributive law, it also equals -3*5 + (-3)*(-5). We know the first thing, (-3*5) equals -15 because of the law in (4). So (-3)*(-5) is doing the job of the additive inverse of -15. We know -15 has exactly one additive inverse, namely 15. Therefore, (-3)*(-5) = 15. A Proof Let a and b be any two real numbers. Consider the number x defined by x = ab + (-a)(b) + (-a)(-b). We can write x = ab + (-a)[ (b) + (-b) ] (factor out -a) = ab + (-a)(0) = ab + 0 = ab Also, x = [ a + (-a) ]b + (-a)(-b) (factor out b) = 0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b) So we have       x = ab and       x = (-a)(-b) Hence, by the transitivity of equality (if a = b, and a = c, then b = c), we have       ab = (-a)(-b)

✉️ Why negative times a negative is a positive? – 🥇 Scalar Scientific Calculator App, Charts & Scripts

Surely everyone knows that the result of multiplying two negative numbers is positive. The formula “minus times a minus is a plus” or “negative times a negative is a positive” was put into our heads during the early school years. However, the teachers have forgotten to explain why this is the case, and to pass the motivation of mathematicians who defined the arithmetic of negative numbers.

⭐️ Multiplication as a short notation of repeated addition

It is said that multiplication is a short notation of repeated addition, which is absolutely true and, with a limitation to integers, a fairly obvious fact.

4 × 3 = 3 + 3 + 3 + 3 = 4 + 4 + 4 = 3 × 4 = 12

⭐️ The commutative property of multiplication and the distributive property of multiplication over addition

These two fundamental multiplication properties can be written as follows

commutative property: a × b = b × a

example: 3 × 4 = 4 × 3

distributive property: a × (b + c) = a × b + a × c

example: 3 × 4 = 3 × (1 + 3) = 3 × 1 +3 × 3 = 3 + 9 = 12

⭐️ Negative numbers multiplication from the mathematician point of view

Mathematicians, defining the arithmetic of negative numbers, wanted to be consistent with the already developed arithmetic of positive numbers and zero. Based on the interpretation of the short notation of repeated addition, we can easily justify the following:

−3 × 4 = (−3) + (−3) + (−3) + (−3) = −12

“By adding debt to debt,” we get more debt – intuitive. Now using the
commutative property of multiplication we get:

4 × (−3) = −3 × 4 = −12

At this point, the intuition is a bit more difficult, but the consistency has been preserved. Time to go to the main point – let’s try to answer the question:

−3 × (−4) = ?

To solve the above, we will use a trick based on the distributive property of multiplication over addition.

−3 × 0 = 0

−3 × 0 = −3 × (−4 + 4) = 0

−3 × (−4 + 4) = −3 × (−4) + (−3) × 4 = 0

−3 × (−4) + (−12) = 0

−3 × (−4) = 12

The above has nothing to do with intuition, but it is consistent, i.e. based on the arithmetic of positive numbers and zero, the commutative property of multiplication, the distributive property of multiplication over addition, we are able to justify why multiplying negative numbers must be a positive number.

⭐️ Multiplication of negative numbers as a reduction of loss

Suppose we multiply two numbers, where the interpretation of the first is the value of profit or losses, while the meaning of the second one is the multiplication (increase / decrease) of the first value. In this situation, multiplying two negative numbers means reducing the loss, ie the total positive effect of the action.

profit × increase = positive effect

profit × decrease = negative effect

loss × increase = negative effect

loss × decrease = positive effect

✅ The above explanation can be described as intuitive 🙂

Finally a video from Mathologer explaining the above problem (that was a base for the above post).

? Negative times a negative in Scalar Calculator? 🙂

Let’s run the negative numbers multiplication test using the Scalar calculator.

Test fully successful 🙂 Thank you for reading! All the best 🙂

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Negative power of numbers and fractions

In this article we will tell you how to raise a number to a negative power and what you can do with it later. What are the properties of negative exponents and what actions can be performed with negative exponents.

What is the power of a number

In mathematics textbooks, you can find the following definition:

“The power n of the number and is the product of factors of and n times in a row”

For example, a ⁿ is the degree, where:

• a is the base of the degree,
• n is the exponent.

Accordingly:

An expression such as a to the power of n is read.

To put it simply, the degree, or rather the exponent (n), tells us how many times a given number (the base of the degree) should be multiplied by itself.

So, if we have a problem where they ask how to raise a number to a power, for example, the number 2 to the third power, then it is solved quite simply:

2 ³ = 2 2 2 where:

• 2 is the base of the power,
• 3 is the exponent.

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Power with a negative exponent

A number to a minus power is equal to a fraction, the numerator of which is one, and the denominator is this number with a positive exponent:

For example, 4 to the minus 2 power is 1/4 ² , 2 to the minus 3 power is 1/2 ³ , 3 to the minus 1 power is 1/3, 10 to the minus one degrees is 1/10 or 0.1.

Examples

Negative exponents help to compactly record extremely small or constantly decreasing values. For example, one part in a billion (0.000000001) can be written as 10 minus 9degrees (10 ^-9 ). In the school curriculum, such values ​​\u200b\u200bare rare: most often they use 10 to the minus 1 degree or 2 to the minus 1 degree.

To figure out how to raise a number to a negative power, let’s remember the rule for dividing powers with the same bases.

The division of degrees with the same bases, but different indicators is carried out according to the following formula: the indicators are subtracted, and the base remains unchanged.

Therefore, if the degree of the dividend is less than the degree of the divisor, then the result will be a number with a negative degree:

a ³ : a ⁶ = a ^{3 — 6} = a ^-3

Operations with negative powers

Multiplication of negative powers

When multiplying negative powers with the same base, the exponents are added together, just as when multiplying positive powers:

a ^m a ⁿ = a ^{m + n}

Examples

Dividing negative powers

When dividing negative powers with the same base, the divisor is subtracted from the exponent of the dividend, just as when dividing positive powers:

Examples 

Raising a fraction to a negative power

To raise a fraction to a negative power, you need to raise the numerator and denominator separately to this power: degree each product factor separately:

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negative number

The negative number is a number that indicates the opposite. If a positive number is the distance up, then a negative number is the distance down. If a positive number is the distance to the right, then a negative number is the distance to the left. If a positive number is a deposit to a bank account, then a negative number is a withdrawal from that bank account. If a positive number is the number of minutes in the future, then a negative number is the number of minutes in the past. If a positive number means addition, then a negative number means subtraction.

Counting numbers (1, 2, 3, etc.) are all positive numbers. Positive numbers, negative numbers, and zero taken together are called «signed numbers» or integers.

The number zero is neither positive nor negative. Zero is its own opposite, i.e. +0 = -0. That is, a zero step to the right is the same as a zero step to the left.

A negative number is always less than zero.

A negative number is written by placing a minus sign «-» before the positive number. For example, 3 is positive and -3 is negative. It is read as «negative three» or «minus three»; it means the opposite of sign 3.

Negative numbers in the number line leave zero. A number and its opposite are always at the same distance from zero. The negative number -3 is to the left of zero, and 3 to the right of zero:

Sometimes we write a pair of opposite numbers as -3 and +3 for emphasis.

A number and its opposite always add up to zero. So the sum of -3 and +3 is 0. You can write this either as -3 + 3 = 0 or as 3 + (- 3) = 0. Also, the number and its opposite are said to «annul each other» .

Using negative numbers

When a person is poor, people sometimes say that he has a negative amount of money. Negative numbers are used in accounting and science.

arithmetic with negative numbers

• Adding a negative number to something is the same as subtracting a positive number from it. For example, adding the negative number «-1» to the number «9» is equivalent to subtracting one from nine. In characters:

9+ (−1) = 9 − 1 = 8

• Subtracting a negative number from something is the same as adding a positive number to it. For example, subtracting the negative number «-8» from the number «6» is equivalent to adding the number «6» and the number «8». In symbols:

6 − (−8) = 6 + 8 = 14

• A negative number multiplied by another negative number gives a positive number. For example, multiplying a negative number «-3» by a negative number «-2» is the same as multiplying a number «3» by a number «2». In characters:

(−3) × (−2) = 3 × 2 = 6

• A negative number multiplied by a positive number gives a negative number. For example, multiplying the negative number «-4» by the positive number «5» is the same as multiplying the number «4» by the number «5», but the answer is no. In characters:

(−4) × 5 = −(4 × 5) = −20

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