Now we will figure out how to perform **subtraction of integers**. First we introduce the terms and notation. Next, we will make sense of the subtraction of integers, from which we proceed to the rule that allows us to reduce the subtraction of integers to addition, and consider examples of using this rule when subtracting a positive integer, negative integer, and zero. After that, we will learn how to check the calculated difference, and see what constitutes the subtraction of integers on the coordinate line.

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## Terms and Notation

To describe the subtraction of integers, we will use all the terms and notation that we used to describe the subtraction of natural numbers.

The integer from which the subtraction is carried out will be called **diminished**. The integer that we subtract will be called **deductible**. The result of the subtraction will be called **difference**.

To denote the subtraction, we will use the minus sign, which we will place between the diminished and subtracted. The reducible, subtracted, and obtained difference will be written in the form of equality. For example, if subtracting the integer b from the integer a, we get the number c, then we can write an equality of the form a − b = c. For example, in an equality of the form −5 - (- 43) = 38, the integer −5 is reducible, the integer −43 is subtractable, and 38 is the difference.

Expressions of the form a − b will also be called the difference, as well as the value of this expression.

Further from the meaning of subtracting integers, it will be understood that the result of subtracting integers is an integer.

## The meaning of subtracting integers

When we studied the subtraction of natural numbers, a connection was established between addition and subtraction, which allowed us to define the subtraction as finding one of the terms by the known sum and the other term. We assume that subtracting integers has the same meaning: **for a given amount and one of the terms is another term** (here anyway you need to know what the addition of integers is).

The stated sense of subtracting integers allows us to state that the difference c − b is equal to a and the difference c − a is equal to b if the sum a + b is equal to c, where a, b and c are integers.

Here are a few examples for specifics.

Suppose that we know that −4 + 9 = 5, then the difference 5−9 is −4. Another example. Suppose we know that the sum of two integers −17 and −3 is −20, then subtracting the integer −3 from the integer −20 gives −17, and the difference −20 - (- 17) is −3.

## The rule for subtracting integers

The meaning of the subtraction of integers, clarified in the previous paragraph, does not give us a way to calculate the difference. Indeed, based on the meaning of subtracting integers, we can only say that one of the known terms is the result of subtracting the other known term from their sum. However, if one of the terms is unknown, then we do not know what the difference between the sum and the known term is equal to. Thus, we need a rule that allows us to subtract another from one integer.

We give the wording **rules for subtracting integers**, after which we give its justification.

To calculate the difference of two integers, add the opposite of the subtracted number to the decremented one, that is, a − b = a + (- b), where a and b are integers, b and −b are opposite numbers.

We prove the stated rule of subtraction, that is, we prove that the value of the expression a + (- b) is equal to the difference between the integers a and b. To do this, by virtue of the meaning of subtracting integers, you need to add subtractable b to a + (- b) and make sure that you get a diminished a, that is, you need to verify the equality (a + (- b)) + b = a. This allows us to do the properties of adding integers, on their basis we can write a chain of equalities of the form (a + (- b)) + b = a + ((- b) + b) = a + 0 = a, which serves as a proof of the subtraction rule integers.

It remains to consider the application of the rule for subtracting integers in solving examples.

## Examples of addition and subtraction of integers

The first thing to learn is to add and subtract integers using the coordinate line. It’s not necessary to draw a coordinate line. It is enough to imagine it in your thoughts and see where the negative numbers are and where the positive ones are.

Consider the simplest expression: 1 + 3. The value of this expression is 4:

This example can be understood using the coordinate line. To do this, from the point where the number 1 is located, you need to move to the right by three steps. As a result, we will be at the point where the number 4 is located. In the figure you can see how this happens:

The plus sign in the expression 1 + 3 tells us that we should move to the right in the direction of increasing numbers.

**Example 2** Find the value of the expression 1 - 3.

The value of this expression is −2

This example can again be understood using the coordinate line. To do this, from the point where the number 1 is located, you need to move left three steps. As a result, we will be at the point where the negative number −2 is located. In the figure you can see how this happens:

The minus sign in the expression 1-3 indicates to us that we should move to the left in the direction of decreasing numbers.

In general, it must be remembered that if addition is carried out, then it is necessary to move to the right in the direction of increase. If subtraction is carried out, then you need to move left to the side of decrease.

**Example 3** Find the value of the expression −2 + 4

The value of this expression is 2

This example can again be understood using the coordinate line. To do this, from the point where the negative number −2 is located, you need to move to the right by four steps. As a result, we will be at the point where the positive number 2 is located.

It can be seen that we moved from the point where the negative number −2 is located to the right side by four steps, and ended up at the point where the positive number 2 is located.

The plus sign in the expression −2 + 4 indicates to us that we should move to the right in the direction of increasing numbers.

**Example 4** Find the value of the expression −1 - 3

The value of this expression is −4

This example can again be solved using the coordinate line. To do this, from the point where the negative number −1 is located, you need to move left three steps. As a result, we will be at the point where the negative number −4 is located

It can be seen that we moved from the point where the negative number −1 is located to the left side by three steps, and ended up at the point where the negative number −4 is located.

The minus sign in the expression −1 - 3 tells us that we should move to the left in the direction of decreasing numbers.

**Example 5** Find the value of the expression −2 + 2

The value of this expression is 0

This example can be solved using the coordinate line. To do this, from the point where the negative number −2 is located, you need to move right two steps. As a result, we will be at the point where the number 0 is located

It can be seen that we moved from the point where the negative number −2 is located to the right side by two steps and ended up at the point where the number 0 is located.

The plus sign in the expression −2 + 2 indicates to us that we should move to the right in the direction of increasing numbers.

## The rules for adding and subtracting integers

To add or subtract integers, it’s not at all necessary to imagine a coordinate line each time, and moreover, draw it. It’s more convenient to use ready-made rules.

Applying the rules, you need to pay attention to the sign of the operation and the signs of the numbers that need to be added or subtracted. This will determine which rule to apply.

**Example 1** Find the value of the expression −2 + 5

Here, a positive number is added to the negative number. In other words, the addition of numbers with different signs is carried out. −2 is a negative number and 5 is a positive number. For such cases, the following rule applies:

**To add numbers with different signs, you need to subtract the smaller module from the larger module, and put the sign of the number whose module is larger before the answer.**

So, let's see which module is larger:

The modulus of 5 is greater than the modulus of −2. A rule requires subtracting a smaller one from a larger module. Therefore, we must subtract 2 from 5, and put the sign of the number whose modulus is larger before the answer received.

Number 5 has a larger module, so the sign of this number will be in the answer. That is, the answer will be positive:

Usually written shorter: −2 + 5 = 3

**Example 2** Find the value of the expression 3 + (−2)

Here, as in the previous example, the addition of numbers with different signs is carried out. 3 is a positive number, and −2 is a negative number. Note that the number −2 is enclosed in brackets to make the expression clearer. This expression is much easier to read than the expression 3 + −2.

So, apply the rule of adding numbers with different signs. As in the previous example, we subtract the smaller module from the larger module and put the sign of the number whose module is larger before the answer:

3 + (−2) = |3| − |−2| = 3 − 2 = 1

The modulus of 3 is greater than the modulus of −2, so we subtracted 2 from 3, and put the sign of that number, the larger modulus, in front of the answer. Number 3 has a larger module, so the sign of this number is put in the answer. That is, the answer is yes.

Usually written shorter than 3 + (−2) = 1

**Example 3** Find the value of expression 3 - 7

In this expression, the larger is subtracted from the smaller number. For this case, the following rule applies:

**In order to subtract the larger from the smaller number, you need to subtract the smaller from the larger number, and put a minus in front of the received answer.**

There is a little snag in this expression. Recall that the equal sign (=) is placed between quantities and expressions when they are equal to each other.

The value of the expression 3 - 7 as we learned is −4. This means that any transformations that we will make in this expression should be equal to −4

But we see that at the second stage the expression 7 - 3 is located, which is not equal to −4.

To correct this situation, expression 7 - 3 must be taken in brackets and put a minus in front of this bracket:

3 − 7 = − (7 − 3) = − (4) = −4

In this case, equality will be observed at each stage:

After the expression is calculated, the brackets can be removed, which we did.

Therefore, to be more precise, the solution should look like this:

3 − 7 = − (7 − 3) = − (4) = − 4

This rule can be written using variables. It will look like this:

**a - b = - (b - a)**

A large number of brackets and signs of operations can complicate the solution of a seemingly quite simple task, therefore it is more advisable to learn to write such examples shortly, for example 3 - 7 = - 4.

In fact, adding and subtracting integers comes down to addition. This means that if you want to subtract numbers, this operation can be replaced by addition.

So, get acquainted with the new rule:

**Subtracting one number from another means adding to the decremented number that is the opposite of the subtracted one.**

For example, consider the simplest expression 5 - 3. At the initial stages of the study of mathematics, we put an equal sign and wrote down the answer:

But now we are progressing in the study, so we need to adapt to the new rules. The new rule says that subtracting one number from another means adding to the decremented number that is opposite to the subtracted one.

Using the example of expression 5 - 3, we will try to understand this rule. The one reduced in this expression is 5, and the subtracted one is 3. The rule says that in order to subtract 3 from 5, you need to add to 5 a number that will be opposite 3. The opposite for number 3 is the number −3. We write a new expression:

And we already know how to find values for such expressions. This is the addition of numbers with different signs, which we examined earlier. To add numbers with different signs, we subtract the smaller module from the larger module, and put the sign of the number whose module is larger before the answer:

5 + (−3) = |5| − |−3| = 5 − 3 = 2

The modulus of 5 is greater than the modulus of −3. Therefore, we subtracted 3 from 5 and got 2. The number 5 has a larger module, so we put the sign of this number in the answer. That is, the answer is positive.

At first, not everyone succeeds in quickly replacing subtraction by addition. This is due to the fact that positive numbers are written without a plus sign.

For example, in the expression 3 - 1, the minus sign indicating subtraction is the operation sign and does not apply to unity. The unit in this case is a positive number, and it has its own plus sign, but we do not see it, since the plus in front of positive numbers is not written down.

And so, for clarity, this expression can be written as follows:

For convenience, numbers with their signs are enclosed in brackets. In this case, replacing subtraction by addition is much simpler.

In the expression (+3) - (+1) in subtracted, this number is (+1), and the opposite number is (−1).

We replace the subtraction by addition and instead of the subtracted (+1) write the number opposite to it (−1)

Further calculation is not difficult.

(+3) − (+1) = (+3) + (−1) = |3| − |−1| = 3 − 1 = 2

At first glance it will seem what is the point of these extra gestures, if you can put an equal sign on the good old method and immediately write down the answer 2. In fact, this rule will help us out more than once.

We solve the previous example 3 - 7 using the subtraction rule. First, we bring the expression to an understandable form, placing each number with its own signs.

The triple has a plus sign, since it is a positive number. The minus indicating the subtraction does not apply to the seven. The seven has a plus sign, since it is a positive number:

Replace the subtraction by addition:

Further calculation is not difficult:

**Example 7** Find the value of the expression −4 - 5

Let's bring the expression to an understandable form:

Before us is again a subtraction operation. This operation must be replaced by addition. To the decremented (−4), we add the number opposite to the subtracted (+5). The opposite number for the subtracted (+5) is the number (−5).

We came to a situation where you need to add the negative numbers. For such cases, the following rule applies:

**To add negative numbers, you need to add their modules, and put a minus in front of the answer received.**

So, we add up the modules of numbers, as the rule requires of us, and put a minus in front of the received answer:

(−4) − (+5) = (−4) + (−5) = |−4| + |−5| = 4 + 5 = −9

The record with the modules must be enclosed in brackets and put a minus in front of these brackets. So we will provide a minus, which should face the answer:

(−4) − (+5) = (−4) + (−5) = −(|−4| + |−5|) = −(4 + 5) = −(9) = −9

The solution for this example can be written shorter:

**Example 8** Find the value of the expression −3 - 5 - 7 - 9

We bring the expression to an understandable form. Here, all numbers except −3 are positive, so they will have plus signs:

Replace subtraction with additions. All the minuses, except for the minus in front of the three, will be exchanged for pluses, and all positive numbers will be reversed:

Now apply the rule of adding negative numbers. To add negative numbers, you need to add their modules and put a minus in front of the answer:

= −( |−3| + |−5| + |−7| + |−9| ) = −(3 + 5 + 7 + 9) = −(24) = −24

The solution to this example can be written shorter:

−3 − 5 − 7 − 9 = −(3 + 5 + 7 + 9) = −24

**Example 9** Find the value of the expression −10 + 6 - 15 + 11 - 7

Let's bring the expression to an understandable form:

There are two operations at once: addition and subtraction. Addition is left unchanged, and the subtraction is replaced by addition:

(−10) + (+6) − (+15) + (+11) − (+7) = (−10) + (+6) + (−15) + (+11) + (−7)

Following the order of actions, we will perform each action in turn, based on the previously studied rules. Records with modules can be skipped:

**First action:**

(−10) + (+6) = − (10 − 6) = − (4) = − 4

**Second action:**

(−4) + (−15) = − (4 + 15) = − (19) = − 19

**Third action:**

(−19) + (+11) = − (19 − 11) = − (8) = −8

**The fourth action:**

(−8) + (−7) = − (8 + 7) = − (15) = − 15

Thus, the value of the expression −10 + 6 - 15 + 11 - 7 is −15

**Note**. Casting an expression into parentheses is optional. When you get used to negative numbers, you can skip this action because it takes time and can be confusing.

So, to add and subtract integers, you need to remember the following rules:

**To add numbers with different signs, you need to subtract the smaller module from the larger module, and put the sign of the number whose module is larger before the answer.**

**To subtract the larger from the smaller number, you need to subtract the smaller from the larger number and put a minus in front of the received answer.**

**Subtracting one number from another means adding to the decremented number that is the opposite of the subtracted one.**

**To add negative numbers, you need to add their modules, and put a minus in front of the answer received.**

## 1 Addition and subtraction of fractions with identical denominators

To add fractions with identical denominators, you need to add their numerators, and leave the denominator the same, for example:

To subtract fractions with the same denominators, subtract the numerator of the second fraction from the numerator of the first fraction, and leave the denominator the same, for example:

To add mixed fractions, you must separately add their whole parts, and then add their fractional parts, and record the result with a mixed fraction,

If the addition of fractional parts resulted in an incorrect fraction, select the integer part from it and add it to the integer part, for example:

## 2 Addition and subtraction of fractions with different denominators.

In order to add or subtract fractions with different denominators, you must first bring them to one denominator, and then proceed as indicated at the beginning of this article. The common denominator of several fractions is the NOC (smallest common multiple). For the numerator of each of the fractions, additional factors are found by dividing the NOC by the denominator of this fraction. We will look at an example later, after we figure out what the NOC is.

## 3 Least common multiple (NLC)

Наименьшее общее кратное двух чисел (НОК) — это наименьшее натуральное число, которое делится на оба эти числа без остатка. Иногда НОК можно подобрать устно, но чаще, особенно при работе с большими числами, приходится находить НОК письменно, с помощью следующего алгоритма:

**Для того, чтобы найти НОК нескольких чисел, нужно:**

- Разложить эти числа на простые множители
- Взять самое большое разложение, и записать эти числа в виде произведения
- Select in other expansions numbers that are not found in the largest expansion (or occur in it a smaller number of times), and add them to the product.
- Multiply all the numbers in the product, this will be the NOC.

**For example, we find the NOC of the numbers 28 and 21:**

## 4 Reduction of fractions to one denominator

Let's get back to adding fractions with different denominators.

When we reduce fractions to the same denominator, equal to the NOC of both denominators, we must multiply the numerators of these fractions by **additional factors**. You can find them by dividing the NOC by the denominator of the corresponding fraction, for example:

Thus, in order to bring fractions to one indicator, you must first find the NOC (that is, the smallest number that is divided by both denominators) of the denominators of these fractions, then put additional factors to the numerators of the fractions. You can find them by dividing the common denominator (LCN) by the denominator of the corresponding fraction. Then you need to multiply the numerator of each fraction by an additional factor, and put the NOC as the denominator.

### Subtraction of a positive integer, examples

Subtract from number 16 a positive integer 36.

According to the rule, to subtract a positive integer 36 from a given number 16, add −36 opposite to the subtracted 36 to the minus 16. That is, the desired difference is equal to the sum of the integers 16 and −36. It remains only to calculate this sum of integers with opposite signs, it turns out to be −20. Thus, the result of subtracting from 16 the number 36 is the number −20.

The whole solution can be written in one line: 16−36 = 16 + (- 36) = - 20.

Subtract −100 from the negative integer.

In order to perform the required action, you need to add the number −50 to the decremented −100, which is opposite to the subtracted 50, - this requires the rule of subtracting integers. Finding the sum of negative integers −100 and −50 should not cause difficulties: −100 + (- 50) = - 150. Therefore, the desired difference is −150.

Briefly finding the difference of the indicated integers can be written as follows: −100−50 = −100 + (- 50) = - 150.

### Subtract Zero Examples

The rule for subtracting integers allows you to get an important result regarding the subtraction of zero from a given integer - **subtracting zero from any integer does not change this number**, i.e, **a − 0 = a** where a is an arbitrary integer.

According to the rule of subtracting integers, subtracting zero is the addition of the opposite number to the decremented number. And since zero is the number opposite to itself, subtracting zero is the same as adding zero. But due to the corresponding property of addition, adding zero to any integer does not change this number. Thus, a − 0 = a + (- 0) = a + 0 = a.

Consider a few examples of subtracting zero from various integers. The difference 45−0 is 45. If we subtract zero from a negative integer −6 005, we obtain −6 005. If zero is subtracted from zero, then as a result we get zero.

### Subtraction of a negative integer, examples

Subtract a negative integer −411 from the integer 0.

The calculation of the difference 0 - (- 411) according to the rule of subtraction of integers reduces to adding to the decremented 0 the number opposite to the subtracted −411. Since the number 411 is opposite to the negative integer −411, then 0 - (- 411) = 0 + 411 = 411.

Calculate the difference −5 - (- 45).

We need to subtract from −5 the integer negative number −45. To do this, we need to calculate the sum of two numbers: minus −5 and number 45, the opposite of minus −45. We have −5 - (- 45) = - 5 + 45 = 40.

### Subtraction of equal integers

I would also like to say about the subtraction of equal integers. The fact is that **if the minus and minus are equal, then their difference is zero**, i.e, **a − a = 0** where a is any integer.

Let us explain the last statement. By the rule of subtraction of integers a − a = a + (- a) = 0. That is, subtracting an equal number from an integer is the same as adding to the given number the opposite number, which gives zero.

Here are a couple of examples. The difference of equal integers −67 and −67 is equal to zero, if we subtract the equal number 653 from 653, then we also get 0. Finally, if we subtract zero from zero, then we get zero.

## Checking the result of subtracting integers

**Checking the result of subtracting integers** carried out by addition. To check whether the subtraction of integers was carried out correctly, it is necessary to add the subtracted to the resulting difference, while the result should be minuscule.

A negative integer −255 was subtracted from the negative integer −303, and the difference −47 was obtained. Is the subtraction correct?

Run a check. To do this, add the subtracted to the difference: −47 + (- 255) = - 302. Since we got a number different from decreasing −303, an error was made somewhere in subtracting integers.