 After learning to add, subtract, multiply, and divide for positive and negative numbers, we need to be able to calculate mixed equations for each.

Addition, subtraction, multiplication, and division are collectively called four rules.

In mathematics, we study the four rules of arithmetic (four arithmetic operations). However, after learning about positive and negative numbers, we have to understand the rules for calculating them in advance. In math calculations, there is some correct order of solving a question that needs to be remembered.

So how should we calculate the four arithmetic operations of positive and negative numbers? We will explain this, including the rules of powers, priority and parentheses.

## Learn the Important Rules in Four Arithmetic Operations

In order to do something, we need to learn the rules. For example, when you play football, you are not allowed to hold the ball in your hands. This is because such rules are set. Because people follow the rules, they will be able to play fair.

It is also because people obey the law that the number of crimes can be reduced. If there are no laws (rules) where looting and robbery are punishable, many crimes will occur.

Similarly, there are rules in mathematics. By following the rules, we can calculate correctly. Some of the rules we need to recognize in four rules of arithmetic are that we must calculate in the following order.

• Calculating powers.
• Calculate the inside of the parentheses.
• Do multiplication and division.

Only in this order can we get the correct answer in mathematics. For example, in the following, we just changed the place of the parenthesis, but the answer is not the same.

• $(2×3)+4=10$
• $2×(3+4)=14$

Changing the position and type of symbols can lead to different answers. Four rules are important in all sorts of situations, including high school and college math, bookkeeping, and programming. So let’s learn the math rules.

### Reasons to Calculate the Power First

The first thing to consider in four rules of arithmetic is powers. If there are powers in the equation, they must be calculated first. Why do we have to calculate the powers first? Because it is a single number. If a power is in an equation, it is impossible to give an answer without first calculating the power.

In mathematics, a calculation cannot be made without revealing definite numbers. For example, calculate the powers first, as follows.

• $2×\textcolor{red}{4^2}=2×\textcolor{red}{(4×4)}=2×\textcolor{red}{16}=32$

The answer to 42 is 16. So, we must replace 42 with 16 beforehand. Since $4^2=16$, this calculation must be done first.

On the other hand, if we ignore the power calculation and try to multiply the numbers by each other, we will get the wrong answer. For example, the following calculation is wrong.

• $\textcolor{red}{2×4}^2=\textcolor{red}{8}^2=8×8=64$

Even though 42 is a single number, if you multiply or divide it by another number without calculating the powers, you will get the wrong answer. For this reason, if there are powers in the equation, we must calculate them first.

### Calculate the Inside of Parenthesis

There are many situations in mathematics where parentheses are used. The important thing is that we must calculate with priority on the inside of the parentheses.

The equation in parentheses is considered to be a number. Although equations are written in parentheses, they must be calculated first because they are regarded as a single number. We explained about powers, that they must be calculated first because they are considered a single number. The same is true for the calculation inside the parentheses. As long as it is a single number, the number in parentheses must be calculated first, otherwise the answer cannot be found.

In mathematics, we use parentheses to deal with the same thing. For example, let’s try to solve the following question.

• $2×\textcolor{red}{(7-3)}=2×\textcolor{red}{4}=8$

You can get the correct answer by calculating $(7-3)=4$ first. On the other hand, it is wrong to ignore the parentheses and calculate the multiplication first, as shown below.

• $\textcolor{red}{2×(7}-3)=14-3=11$

If there are parentheses, the calculation within the parentheses must be done first.

In addition, sometimes parentheses can contain powers. In this case, we can calculate the power and then calculate the inside of the parentheses. For example, the following.

• $2×\textcolor{red}{(7-3^2)}=2×\textcolor{red}{(7-9)}=2×\textcolor{red}{-2}=-4$

Since powers are a single number, it is impossible to calculate in parentheses without calculating powers. Therefore, understand that the calculation of powers must be done first.

### Multiplication and Division Are Calculated Before Addition and Subtraction

The next calculation to be done is multiplication and division. Multiply and divide first, and additions and subtractions are done later. Why is multiplication (or division) a priority? It’s because you have to take units into account.

When calculating in math, many questions do not have units. But in real, everyday life, units do exist. Because mathematics is used in everyday life, we have to be aware of units. The equations just omit the units and the units are hidden.

In those cases, there are only a few situations in our daily lives where we have to add or subtract. This is because in order to add or subtract, the units must be the same. For example, if you add 3 kg and 5 kg, you get 8 kg. But you can’t add 3 kg and \$2. This is because the units are different. The important fact is that we are not allowed to add or subtract at will. On the other hand, multiplication (or division) can be calculated, even if the units are different. For example, if you buy 3kg of bananas that cost \$2.00 per kilogram, it would be as follows.

• $\$2 × 3kg = \$6$

Although multiplication uses kg in the equation, the unit of the answer is dollars. In multiplication (or division), we can change the units.

As mentioned above, we can add or subtract in the same units. The reason we must do multiplication and division before addition and subtraction is because we must have the same units.

Make sure to understand these rules and calculate them. After calculating powers, parentheses, multiplication, and division, the last step is addition and subtraction.

You can add and subtract after you have calculated all the numbers and the units are correct. Understand that there are these rules in mathematics.

## Calculating from Left to Right? Convert Division to Fraction Multiplication

So far, we have explained the order of precedence and the reasons for calculating plus and minus numbers (four arithmetic operations). However, some people may be taught the rule of calculating from left to right in addition to the previous explanation. However, it’s okay to forget this rule.

For addition and multiplication, the answer is the same, no matter how you calculate it. As mentioned above, you must follow the rules about multiplication that must take precedence over addition and subtraction. However, whether you calculate from the left or the right, the answer is the same.

Nevertheless, why do we have to calculate positive and negative numbers in the four rules, starting from the left? That’s because it includes division. When division is included in the equation, we cannot get the correct answer unless we calculate it from the left side.

For example, let’s do the following calculation.

• $\textcolor{red}{6÷2}×3=\textcolor{red}{3}×3=9$

Calculating from left to right, the answer is 9. On the other hand, if we calculate from right to left, what happens? The result is as follows.

• $6÷\textcolor{red}{2×3}=6÷\textcolor{red}{6}=1$

As you can see, the answer depends on whether you are calculating from the left or the right. If you check the equation, you will see that even though it must be divided by 2, the wrong calculation shows that it is divided by 6. Because of this calculation error, we are generally told to calculate from the left to the right.

### Don’t Use Division and Correcting All Fractions Multiplication

But in fact, division is inconvenient in mathematics. There are so many cases where it cannot be divided and it is not useful. Therefore, in practice, division is rarely used in mathematics above junior high school, including high school and college.

Fractions are used as an alternative to division. Division can be substituted for multiplication of fractions as follows. Division is used in elementary school math. But after studying plus and minus numbers in math, consider that we don’t use division in four rules of arithmetic. All of us convert division into multiplication of fractions.

Therefore, for the previous calculation, we should correct and calculate as follows.

• $6\textcolor{red}{÷2}×3=6\textcolor{red}{×\displaystyle\frac{1}{2}}×3=9$

With multiplication, the answer is the same whether you calculate from the left or the right. And even if we cannot divide, we can use fractions to get the answer. Dividing decimals is also easy to calculate by multiplying by fractions.

Division is not useful and is rarely used in math calculations. So be sure to change division to multiplication of fractions.

## Exercise: Four Arithmetic Operations of Plus and Minus Number

Q1. Do the following calculation.

1. $-5-4×-3$
2. $3+8÷(-2)$
3. $-2^3×(5-3^2)$

In the four rules of arithmetic of plus and minus numbers, to review, the following order must be used.

1. Calculate the multiplier.
2. Calculate the inside of the parentheses.
3. Do multiplication and division.

Calculated in this order, we get the following.

(a)

$-5-\textcolor{red}{4×-3}$

$=-5-\textcolor{red}{(-12)}$

$=-5+12$

$=7$

(b)

$3+\textcolor{red}{8÷(-2)}$

$=3+\textcolor{red}{(-4)}$

$=3-4$

$=-1$

(c)

$-\textcolor{red}{2^3}×(5-\textcolor{blue}{3^2})$

$=-\textcolor{red}{8}×(5-\textcolor{blue}{9})$

$=-8×(-4)$

$=32$

Q2. Do the following calculation.

1. $3^2-(-3)^2×(-2)^2$
2. $6÷\displaystyle\frac{3}{2}×3+\displaystyle\frac{1}{2}$
3. $(7-3^2)×2^2-(-6)÷(-3)^2$

This is a bit more complicated than the previous calculations. Also, there are some divisions that are not divisible. For the division, as mentioned earlier, you should correct the multiplication of fractions.

Then you get the following.

(a)

$3^2-(-3)^2×(-2)^2$

$=9-9×4$

$=9-36$

$=-27$

(b)

$6\textcolor{red}{÷\displaystyle\frac{3}{2}}×3+\displaystyle\frac{1}{2}$

$=6\textcolor{red}{×\displaystyle\frac{2}{3}}×3+\displaystyle\frac{1}{2}$

$=12+\displaystyle\frac{1}{2}$

$=\displaystyle\frac{24}{2}+\displaystyle\frac{1}{2}$

$=\displaystyle\frac{25}{2}$

(c)

$\textcolor{red}{(7-3^2)×2^2}-\textcolor{blue}{(-6)÷(-3)^2}$

$=\textcolor{red}{(7- 9)×4}-\textcolor{blue}{(-6)÷9}$

$=\textcolor{red}{-2×4}-\textcolor{blue}{(-6)×\displaystyle\frac{1}{9}}$

$=-8-(-\displaystyle\frac{6}{9})$

$=-8-(-\displaystyle\frac{2}{3})$

$=\textcolor{red}{-8}+\displaystyle\frac{2}{3}$

$=\textcolor{red}{-\displaystyle\frac{24}{3}}+\displaystyle\frac{2}{3}$

$=-\displaystyle\frac{22}{3}$

## Four Rules of Arithmetic for Addition, Subtraction, Multiplication and Division

When learning plus and minus numbers, four rules of arithmetic become more complicated. Not only do we have to think about what the signs will be, but we also have to learn rules of four arithmetic operations because we use multipliers and parentheses to calculate them.

We have explained how to calculate four arithmetic operations, including why they are such math rules. There is a correct order for solving questions in mathematics, so be sure to calculate in the correct order.

Also, for division, you can correct it to multiplying fractions. Division is almost never used. In fact, no one uses division when learning more advanced mathematics, such as in high school. Instead, they use multiplication of fractions to calculate. Therefore, you should start getting into the habit of correcting division to multiplication.

There are rules and tricks for solving math questions. Make sure you understand the rules of math, including the reasoning behind them, so that you can solve the questions.