When you learn math, you have to understand negative numbers. Unlike when you were only learning positive integers, learning the concept of zero and addition and subtraction will allow you to perform complex calculations.

It may seem strange to say that there are numbers smaller than zero. However, we use negative numbers in many aspects of our daily lives. Therefore, it is very important to use positive as well as negative integers to calculate.

So what is the concept of 0 and negative numbers? How can we think and calculate positive and negative numbers?

When learning mathematics, we all need to understand positive and negative numbers. This section will explain how to calculate positive and negative integers, with questions.

Table of Contents

## 0 (Zero) Is Used to Indicate a Criteria

In general, 0 (zero) indicates that there is nothing. For example, if you have \$1, if you spend it, the money you have is \$0.

If non-existence means zero, the concept of a negative doesn’t make sense for us. When learning about negative integers, many people get confused because they think that there are numbers below that which do not exist.

However, 0 does not just indicate that it does not exist. It has other meanings as well. It is a criterion. By using zero as a criterion, we use positive and negative numbers, depending on whether the number is greater or less than zero.

For example, we use temperature in our daily lives. Temperature has a standard of 0°C (32°F). If the temperature is lower than 0°C, the number is negative. If a city has snow, many people will experience negative temperatures.

Given this fact, we can realize that it’s not strange to use negative numbers as well as positive numbers; since zero has a meaning of indicating a standard, many people use negative numbers in their daily lives.

### Positive Integers (Natural Numbers) and Negative Integers Indicate Increase or Decrease from Reference

Once you learn that there is a concept of a standard in 0, you will understand what a positive and negative integer means. Positive integers (natural numbers) and negative integers are how far off the reference point (zero) is.

In the following diagram, based on zero, the number to the right is a positive integer (natural number).

On the other hand, a number to the left of zero is a negative integer. The positive and negative numbers can be used to represent an increase or decrease from a standard.

For example, let’s consider losing weight by going on a diet. In that case, the majority of people use their current weight as a reference point (zero). If you have lost 3kg of weight, then you can say that you have successfully gained -3kg of weight.

**-You Can Replace the Positive and the Negative**

Therefore, the number of pluses and minuses can be replaced. For example, the following means the same thing.

- I’ve lost 3kg of weight.
- I’ve gained -3kgof weight.

The following also means the same thing.

- The temperature drops by 5°C.
- The temperature goes up by -5°C.

It is important to understand that these rephrases are possible when learning about positive and negative numbers.

### How to Distinguish Between Large and Small, and Absolute Value

With positive and negative numbers, how can we distinguish between large and small numbers? In order to understand this, we need to understand the concept of absolute value.

We have understood that we can identify positive and negative numbers based on zero. When identifying a number large or small, check how far away from zero it is. Regardless of whether the number is positive or negative, the absolute value is how far off from zero the number is.

For example, the absolute value of 4 is 4. On the other hand, the absolute value of -3 is 3. If you move three steps to the left from 0, you get -3. The distance from 0 is 3, so the absolute value of -3 is 3

Once we learn this concept, we will be able to distinguish between large and small numbers. For example, with 5 and 3, we can see that 5 is a larger number than 3. Therefore, we can show that 5>3. In contrast, which number is larger, -3 or -5?

The absolute value of -3 is 3. Also, the absolute value of -5 is 5. What is important is the fact that the higher the number (absolute value) of negative numbers, the smaller the number. The absolute value of -5 is greater than -3. Therefore, -3 is a larger number and can be expressed as $-3>-5$.

It may seem strange that even though the number is large, a negative number makes it small. However, we use this in our daily lives as well.

For example, it is colder at -20°C (-4°F) than it is at -3°C (26.6°F). This is because -20°C is a larger negative number. Understand that even though the number (absolute value) is large, it will be a small number in a negative number.

## Addition and Subtraction of Positive Numbers

Understanding the concepts of zero and negative numbers will help you understand addition and subtraction for positive and negative numbers.

It is easier to understand if we start out by learning about addition and subtraction of positive numbers. When dealing with positive numbers, there are two types as follows.

- Addition of positive numbers
- Subtraction of positive numbers

We will explain each of these.

### Use Parentheses in the Addition of Positive Numbers

You have already learned about the addition of positive numbers in elementary school math. Therefore, you will be able to understand the addition of positive numbers and positive numbers without any difficulty. For example, the following calculation is easy to do.

- $3+1=4$

In this case, $3+1$ can also be expressed as follows.

- $3+(+1)$
- $(+3)+(+1)$

They all have the same meaning. In addition, you can use two +’s in a row. However, instead of writing ++, we use parentheses.

**-Add a Positive Number to a Negative Number**

How can we add a positive number to a negative number? For example, the following calculation.

- $-5+3$

It can also be expressed as $-5+(+3)$ or $(-5)+(+3)$. In any case, this calculation requires adding 3 to -5. In other words, you must answer 3 higher than -5.

The absolute value of -5 is 5. By adding +3 to -5, three to the right, the answer is -2. Therefore, the answer is -2.

### Rules for Subtracting Positive Numbers

In contrast, how can we think about subtracting positive numbers? For example, suppose we have the following calculation.

- $3-2=1$

We have already learned about this equation in elementary school. It is important to note that it can be replaced by the following equations.

- $3-(+2)=1$
- $(+3)-(+2)=1$

All positive numbers have a + hidden in them. For example, when we use the number 2 or 3, we can represent it as +2 or +3. For positive integers (natural numbers), the + can be omitted. Just to be detailed, $3-2$ can be represented as $3-(+2)$.

**-Subtract a Positive number from a Negative number**

Of course, it is common to subtract positive numbers (natural numbers) from negative numbers. For example, what is the answer to the following calculation?

- $-1-3$

This equation can also be replaced with $-1-(+3)$ or $(-1)-(+3)$. In order to subtract +3 from -1, the answer must be 3 less than -1.

Therefore, the answer is -4.

- $-1-3=-4$
- $-1-(+3)=-4$
- $(-1)-(+3)=-4$

It is calculated like this.

## Addition and Subtraction of Negative Numbers

We’ve discussed positive numbers, and it’s not difficult to understand addition and subtraction for positive numbers. What is at issue is addition and subtraction for negative numbers.

- Adding the negative numbers
- subtract the negative numbers

What do these mean? For example, when we calculate $1 – (-3)$, we get the following.

- $1-(-3)=1+3=4$

Many people try to learn how to calculate without understanding why. However, you have to understand the reason. So we will explain it, including the reasons.

### Learn to Add Negative Numbers by Replacing

Calculations to add negative numbers are frequently given. For example, how should the following calculation be considered?

- $2+(-4)$

As mentioned above, the numbers can be rephrased. For example, if you lose 2kg of weight, you can rephrase it as “I’ve gained -2kg of weight”. Even though it is expressed as an increase in weight, it is a -2 kg plus, so you have lost weight.

In this way, you can replace the positive with the negative. In other words, we can rephrase it as follows.

- $2-4$:4 less.
- $2+(-4)$: -4 higher

Subtract 4 from 2 is -2. A number 4 less than 2 is -2. Understand that adding a negative number is, in essence, the same as subtracting a positive number.

### Why Subtracting a Negative Is a Positive Result.

When understanding negative numbers, it is the subtraction of the negative that is confusing to many people. The following is what we have explained so far.

- Addition of positive numbers (positive integers)
- Subtraction of positive numbers (positive integers)
- Addition of negative numbers (negative integers)

Addition and subtraction of positive numbers are already taught in elementary school. The addition of negative numbers can be thought of as the subtraction of positive numbers (natural numbers). On the other hand, how can we calculate the following, for example?

- $2-(-3)$

We can rephrase this calculation as well. The minus sign has the opposite meaning. As mentioned above, the expression “I’ve lost 2kg” can be reworded as “I’ve gained -2kg”.

So what does it mean if we rephrase the expression -3kg less? Since minus has the opposite property, saying that you have lost -3kg means that you have gained 3kg. The negative of a negative is a positive.

That’s why when you subtract a negative number, you can change it to a positive number.

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

It is important to note that negative has the opposite meaning. A lot of people don’t understand in mathematics why subtracting a negative means a positive. Once you understand that a negative has the opposite meaning and can be rephrased, you will understand this reason.

## Exercises: Addition and Subtraction of Positive and Negative Numbers

**Q1.** Use inequality signs to represent the following numbers in terms of size.

- $-2、4、-3$
- $-0.3、-3、0$

**A1.** Answers.

For negative numbers, the higher the number (absolute value), the smaller the number. Since the concept is the opposite of positive numbers, the order of inequality is as follows.

- $4>-2>-3$
- $0>-0.3>-3$

For example, in (b), when comparing 0, 0.3 and 3, the order of the higher numbers is $3>0.3>0$. For a negative number, the opposite is true, the higher the number (absolute value), the smaller it is, so $0>-0.3>-3$.

**Q2.** Do the following calculation.

- $7-(+3)$
- $-0.5+(-0.2)$
- $-3-(-6)$
- $\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{1}{3}\right)$
- $-3 +(-2)-(-6)-5$

**A2.** Answers.

In calculating positive and negative numbers, let’s rephrase them. In both junior high school math and high school math, everyone calculates after replacing signs. The point of substitution is as follows.

- $+$ and $+$ becomes $+$
- $+$ and $-$ becomes $-$
- $-$ and $+$ becomes $-$
- $-$ and $-$ becomes $+$

In addition and subtraction, both decimals and fractions can all be rephrased in this way. Therefore, we can calculate as follows.

**(a)**

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

**(b)**

$-0.5+(-0.2)=-0.5-0.2=-0.7$

**(c)**

$-3-(-6)=-3+6=3$

**(d)**

$\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{1}{3}\right)$

$=\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}$

$=\displaystyle\frac{3}{6}+\displaystyle\frac{2}{6}=\displaystyle\frac{5}{6}$

**(e)**

$-3+(-2)-(-6)-5$

$=-3-2+6-5=-4$

You can’t solve the problem without rephrasing the positive and negative. Be sure to check to see if the addition or subtraction. Once you change the signs, all you have to do is add or subtract, as this is elementary school math content.

Fractions are a little more complicated because you need to reword the signs and then adjust the denominator. However, calculating fractions should be taught in elementary school, so it is not difficult.

**Q3.** Do the following calculations.

The table below shows how many centimeters taller the four people in A to D are from 160cm.

A | B | C | D | |

Difference from 160cm (cm) | 12 | -4 | 7 | -9 |

- How many centimeters taller would the tallest person be than the shortest person?
- Find the average height of the four people.

**A3.** Answers.

**a.** How many centimeters taller would the tallest person be than the shortest person?

The tallest person is A. And the shortest person is D. A is +12cm taller and D is 9cm shorter.

When calculating the difference between the numbers, you have to subtract. For example, according to the diagram above, A is +12cm tall and C is +7cm tall; the difference in height between A and C is $12cm-(+7cm)=12cm-7cm=5cm$. In the same way, we use subtraction in comparing numbers.

The height difference between A and D is calculated as follows.

- $12-(-9)=12+9=21cm$

Therefore, the tallest person is 21cm taller than the shortest person.

**b.** Find the average height of the four people.

When calculating the average height, one way to calculate the average height is to calculate the respective heights from A to D. Using the table above as a guide, the respective heights are as follows.

A | B | C | D | |

Height (cm) | 172cm | 156cm | 167cm | 151cm |

After adding the heights from A to D, you can divide the heights by four to get the average of the heights. But can’t we find the average in an easier way?

In this regard, try to calculate how much the average is off. The four people are different in height, as shown below.

- $A: +12cm$
- $B: -4cm$
- $C: 7cm$
- $D: -9cn$

What is the average height difference between the four people? If you add up the height difference between the four people, it is as follows.

- $12+(-4)+7+(-9)$
- $=12-4+7-9=6cm$

The total difference in height between the 4 people is 6cm. If you divide the total by the number of people, you can calculate the average value. In other words, we can get the average value for how many centimeters higher (or lower) per person from 160cm.

If you divide 6 cm (total value) by 4 people, you get $6÷4=1.5cm$. This means that the average height of the 4 people is +1.5 cm taller than 160 cm. Therefore, the average height of the 4 people is 161.5 cm.

- $160+1.5=161.5cm$

## Use Positive and Negative Numbers to Add and Subtract

When learning negative numbers, many people are confused; they don’t understand why there are values lower than 0 (zero). And the majority of people, even adults, can’t explain why subtracting a negative number is a positive number.

However, as we’ve explained, we find that we use negative numbers in many situations, even in our daily lives: 0 means a standard value, and we compare the size of a number by its absolute value.

In addition, when we actually use plus and minus numbers to add and subtract, we often use parentheses. Whenever you use parentheses in a calculation, you can replace them. Even when solving difficult problems in high school and college, everyone uses replacement, so be sure to calculate after you’ve done so.

Once you learn these rules, you will be able to solve addition and subtraction questions with positive integers (natural numbers) and negative integers. The method is the same for decimals and fractions. Many people have trouble solving questions, especially with negative subtraction, so make sure you understand including the reasons why.