When learning negative numbers, many people get confused. This is because when you multiply or divide different signs, the signs change. In addition, when you multiply negative by negative, the sign becomes positive. Why does this happen?

Few people can explain this reason. However, to understand the math, we need to resolve the question beforehand.

Once you understand the signs in multiplication, you will be able to solve not only division and powers, but also application questions in four arithmetic operations.

Many people find the calculation of negative numbers difficult because they do not understand the concepts. So, we will explain why the signs change with negative multiplication and how to solve the questions.

## Depending on the Difference in Signals, Multiplication Can Be Plus or Minus

If you study only positive numbers, you don’t have to consider negative numbers. However, in mathematics, you have to deal with negative numbers.

There are multiplication and division in mathematics. When you do multiplication, the sign of the answer changes depending on what sign is used. In multiplication, the result is as follows.

• Positive number $(+)×$ Positive number $(+)=+$
• Negative number $(-)×$ Positive number $(+)=-$
• Positive number $(+)×$ Negative number $(-)=-$
• Negative number $(-)×$ Negative number $(-)=+$

Some might think that we have to remember four of them. However, you don’t have to remember this. It is as follows.

• Multiplication of same signs results in $+$.
• Multiplication of different signs results in $-$.

For example, $4×-3=-12$. This is because it is a multiplication of different signs. On the other hand, $-4×-3=12$. Because it is a multiplication of two negative signs.

But why is the answer negative if we multiply a positive number by a negative number? And why does multiplying a negative number by a negative number result in a positive answer? Few people can explain this reason correctly, so we’ll try to answer this question.

### Why Multiplication by Plus and Minus Is Possible?

When multiplying, why do positive and negative answers become negative? Let’s solve this question first.

In general, when we solve this question, we think in terms of speed, time and distance. The following formula exists.

• Speed $×$ Time $=$ Distance

For example, if you walk at a speed of 1 km/h and 3 hours pass, you will have walked a distance of 3 km. Therefore, we can see that it is possible to multiply positive numbers with each other.

-Why Is the Answer Negative for Different Signs?

So why does the use of different signs make the answer negative? The 0 (zero) represents a reference point. And if you use only positive signs, you are only considering the future. But in reality, the past also exists.

For the distance walked, we can create a diagram like the following.

In this case, if you walk at -1km/h and 3 hours pass, where will you be? Walking at -1 km/h means that you are walking westward at a speed of 1 km/h. If you walk westward and 3 hours pass, you are at -3km.

• $1km/h×3hours=-3km$

Also, if you walk at 1 km/h and -3 hours pass, where will you be? -3 hours have passed means that you were 3 hours ago. If you walk eastward at a speed of 1 km/h, 3 hours ago, you are at -3 km.

• $1 km/h×-3hours=-3 km$

If you think about it this way, you can understand why multiplying a positive number by a negative number results in a negative answer.

### Why Multiplying a Negative Number by a Negative Number Makes a Positive

In contrast, why is multiplying a negative number by a negative number result in a positive number? Let’s consider the same thing as before, in terms of speed, time and distance. If you walk at -1 km/h and -3 hours pass, where will you be?

Walking at -1 km/h means you are walking westward. Also – 3 hours have passed means that 3 hours ago.

If you are walking west (minus), 3 hours ago, you would be at +3 km (3 km to the east).

• $-1km/h×-3hours=+3km$

This way of thinking allows us to see why the answer is positive when we multiply the negative by the negative.

## The Signs Change for Division as well

We have explained multiplication so far. On the other hand, what about division?

We can think of division in exactly the same way as we think of multiplication. That is, for division between same signs, the answer is a positive sign. On the other hand, for division of different signs, the answer is a negative sign.

• Divisions with same signs result in $+$.
• Division with different signs results in $-$.

The reason for this is that both multiplication and division are the same. For example, if you walk at a speed of 1 km/h for 3 hours, you will walk 3 km.

• $1km/h×3hours=3km$

So, if you are walking a distance of 3 km in 3 hours, what is your speed? The answer to this question is as follows

• $3km÷3 hours=1km/h$

Thus, multiplication can be turned into division.

For example, in the division of fractions, you can reverse the top and bottom of a fraction to make the equation converted to multiplication. This is because division can be transformed into multiplication. Multiplication and division have the same properties. Therefore, the change in sign in division is the same as in multiplication.

### The Sign Changes Depending on Whether the Minus is Even or Odd

However, when doing math calculations, it is rare to calculate only two numbers. Often, we multiply or divide multiple numbers. What are the signs when we multiply a number of positive or negative numbers?

For this question, if the negative sign of the multiplication (or division) is an even number, the sign of the answer is positive. In contrast, if the negative sign of the multiplication (or division) is an odd number, the sign of the answer is negative.

 Number of minuses Equation Answer sign 1 (odd) $+×-$ $-$ 2 (even) $-×-$ $+$ 3 (odd) $-×-×-$ $-$ 4 (even) $-×-×-×-$ $+$ 5 (odd) $-×-×-×-×-$ $-$

As mentioned above, the negative has the opposite nature. The opposite of the opposite is positive. Therefore, if the negative is even, the sign of the answer is positive. In contrast, if the negative is odd, the sign of the answer is negative.

Not only integers, but also decimals and fractions follow this rule. For example, the following is true.

• $-1×-1×-\displaystyle\frac{1}{2}=-\displaystyle\frac{1}{2}$

Since the number of minuses is three and odd, the answer is negative. In multiplication and division calculations, the sign of the answer changes depending on whether the number of minuses is even or odd.

## How to Calculate Power Using Exponent

In positive and negative numbers, powers are as important a concept as multiplication and division. Power is a calculation that expresses how much is multiplied by the same number. In powers, for example, $3 × 3$ is expressed as 32. Also, $4 × 4 × 4$ is expressed as 43.

In terms of reading, 32 is 3 squared. 43 is 4 cubed.

• 32 = 3 squared (or 3 to the 2nd power)
• 33 = 3 cubed (or 3 to the 3rd power)
• 34 = 3 to the 4th power
• 35 = 3 to the 5th power
• 36 = 3 to the 6th power

The small number in the upper right corner is called the exponent. For example, for 43, 3 is the exponent. By checking the exponent, you can see how many times you need to multiply the same number.

Calculating the power using exponents, for example, gives us the following.

• $3^4=3×3×3×3=81$
• $(-2)^3=-2×-2×-2=-8$
• $0.1^2=0.1×0.1=0.01$

When calculating powers, be sure to check how many numbers you need to multiply by looking at the exponent.

### Note the Difference in Signs and Parentheses in Calculating Power

Note that recognizing the difference between signs in the calculation of powers is the most important thing when answering the question. For example, what would be the answer to the following power equations?

• $-3^2$
• $(-3)^2$
• $-(-3)^2$

If you don’t understand the difference, you won’t be able to calculate the powers correctly. So, let’s learn the rules of exponents. Exponents have the following properties

• Power to the previous number: the sign (-1) is not included

It is an exponent that does not include the sign and only powers the previous number. Therefore, -32 is calculated as follows.

• $-3^2=-1×3^2=-1×3×3=-9$

There is a -1 hidden in the -32. So let’s separate -1 and 32. The rule is that an exponent is only valid for the previous number and does not include the sign (-1).

However, we just explained earlier that $(-2)^3=-2×-2×-2=-8$. Why is this despite the fact that exponents do not include signs? That’s because it uses parentheses.

In parentheses, the rule is that the equation in parentheses is considered to be a single number. In (-2)3, there is -2 in parentheses, so -2 is a number. If we consider -2 as a number, (-2)3 is “$(-2)^3=-2×-2×-2$”. In the same way, (-3)2 becomes the following.

• $(-3)^2=-3×-3=9$

In contrast, what does -(-3) 2 become? As mentioned above, the exponent is only multiplied by the previous number. Also, the equation in parentheses is considered as a single number. Therefore, the equation is as follows.

• $-(-3)^2=-1×(-3)^2=-1×-3×-3=9$

The exponent to the right of (-3) is 2. Therefore, it must be -3 × -3. However, it has to be calculated separately from the sign (-1) in front of (-3). As a result, you get the calculation as shown above.

### Note the Parentheses Even in the Power of Fractions

Depending on whether or not parentheses are present, the answer to the question varies greatly when calculating powers. It is important to pay attention to the parentheses, even for powers of fractions.

For example, what is the answer to the following question?

• $\displaystyle\frac{2^4}{3}$
• $\left(\displaystyle\frac{2}{3}\right)^4$

The exponent is only valid for the previous number. Therefore, consider the following.

In $\displaystyle\frac{2^4}{3}$, only 2 is to the 4th power. This is because the previous number is 2.

On the other hand, in $\left(\displaystyle\frac{2}{3}\right)^4$, $\left(\displaystyle\frac{2}{3}\right)$ to the 4th power, because there are parentheses in front of the exponent.

If there are no parentheses, then only the number in front of the exponent can be calculated to the power. In contrast, if there are parentheses, we need to multiply all the numbers in parentheses to the power. It must be understood that in calculations using exponents, the method of calculation varies depending on whether parentheses are present or not.

## Exercises: Math Multiplication, Division and Power

Q1. Do the following calculation.

1. $3×-4$
2. $-30÷-5$
3. $-4×-6×-3$

When multiplying or dividing using positive and negative numbers, the important thing to remember is that if the negative number is even, the answer is positive, and if the negative number is odd, the answer is negative. Therefore, the answer is as follows.

1. $3×-4=-12$
2. $-30÷-5=6$
3. $-4×-6×-3=-72$

Q2. Do the following calculation.

1. $-3^2×(-2)^2$
2. $(-3)^3×8÷2^2$
3. $-(-2)^4÷\displaystyle\frac{1}{3}×\left(-\displaystyle\frac{2}{3}\right)^2$

When we solve a math question, we must first calculate powers. This is because powers are the same as a single number. For example, when solving the question $12÷2^2$, it is impossible to solve the question without first calculating 22. 22 is $2×2=4$, so $12÷4=3$.

In the same way, we need to calculate the powers first. Then we get the following.

1. $-3^2×(-2)^2=-9×4=-36$
2. $(-3)^3×8÷2^2=-27×8÷4=-54$

Consider the question in (c) as follows.

$-(-2)^4÷\displaystyle\frac{1}{3}×\left(-\displaystyle\frac{2}{3}\right)^2$

$=-16×3×\displaystyle\frac{4}{9}$

$=-\displaystyle\frac{192}{9}$

$=-\displaystyle\frac{64}{3}$

By calculating each one separately, we can finally make the equation for multiplication only.

## Multiply and Divide to Calculate Plus and Minus

We can’t do math calculations unless we understand how to calculate plus and minus. So when you multiply and divide, learn how the signs work.

So many people are confused, especially when multiplying (or dividing) a negative number by a negative number, because they don’t understand why it’s positive. Therefore, we have explained why the answer is positive when multiplying a negative number by a negative number.

Also important is the fact that if the negative number is an even number, the answer is positive, and if the negative number is an odd number, the answer is negative.

In addition to that, you need to learn the concept of power. Parentheses are important in powers, and the answer depends on where the parentheses are placed. This is the concept of multiplication and division that you must learn with positive and negative numbers.