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# Subtraction

## What is Subtraction?

The process of finding the difference between two numbers is known as subtraction. In other words, it is the process of taking away a number from another. The number that is subtracted is called the subtrahend while the number from which the subtrahend is subtracted is called minuend. The result of this subtraction is called the difference.

## Formula and Symbol for Subtraction

The symbol used for showing subtracting two numbers is ( – ).

The formula for subtraction is

Minuend – Subtrahend = Difference

For example, you had 10 pencils. Your sister borrowed 6 pencils from you. So, how many pencils were you left with? You were left with 4 pencils. How did we get the number 4?

We used subtraction to obtain the number of pencils left with you.

Mathematically, the above problem can be represented as

Number of pencils you had = 10

Number of pencils your sister borrowed from you = 6

Number of pencils left with you = 10 – 6 = 4

Here 10 is the minuend, i.e. the number from which another number is being subtracted.

6 is the subtrahend, i.e. the number that is being subtracted.

Remember:

Subtraction is just the opposite of addition. Also, every subtraction problem can be re-written as an addition problem.

For example,

We just saw that 10 – 6 = 4

This can be rewritten as 6 + 4 = 10 which is an addition problem.

## Graphical Representation of Subtraction

Let us understand subtraction through a graphical representation.

Suppose there were 10 trees in a garden.

One day, six trees fell down due to strong winds.

Count the number of trees that are now left in the garden.

The number is 4.

Hence 10 – 6 = 4

## Subtracting a larger number from a smaller number

Suppose we have two numbers, 3 and 9. We wish to subtract 3 from 9. The difference we get upon the subtraction is 6.

Hence 9 -3 = 6

But, what if we want to do it the other way round? That is, we wish to subtract 9 from 3. Is that possible?

Can we subtract a larger number from a smaller number? Remember the number lines about which you must have studied earlier. They have negative numbers too.

When you subtract a larger number from a smaller number, you subtract their whole values and place a sign of minus (-) against the difference.

This means that we subtract 9 from 3, we actually subtract 3 from 9, where we get the answer as 6 and place the minus sign before it, making the net answer as -6.

Hence, 3 – 9 = -6

## Subtraction of Two-Digit Numbers (Without borrowing)

### Case 1 – Both the subtrahend and the minuend are two-digit numbers

When we subtract a one-digit number from another one-digit number, we directly subtract the subtrahend from the minuend and get the result.

But, in two digits numbers, we have digits at two places, tens and ones. So, in order to subtract one two-digit number from another, we place the subtrahend below the minuend in the order of their place values and subtract accordingly.

The following are the steps involved in subtraction:

1.  Place the values vertically in order of their place values.

2.  Start subtracting the numbers, starting from the one’s place.

For example, let us subtract 43 from 68. Both are two-digit numbers, therefore placing one number below the other in order of their place values, we get –

### Case 2 – The minuend is a two-digit number while the subtrahend is a one-digit number

If a minuend is a two-digit number and the subtrahend is a one-digit number, we place one number below the other in order of their place values.

For example, let us subtract 5 from 79. Here the minuend, 79 is a two-digit number while the subtrahend, 5 is a one-digit number. So, in order to subtract a one-digit number from a two-digit number, we place the subtrahend below the minuend in the order of their place values and subtract accordingly.

## Subtraction of Three-Digit Numbers (Without borrowing)

### Case 1 – Both the subtrahend and the minuend are three-digit numbers

Subtraction of three-digit numbers is in a way similar to that of the two-digit numbers. In three digits numbers, we have digits at three places, hundreds, tens, and ones. So, in order to subtract one three-digit number from another, we place the subtrahend below the minuend in the order of their place values and subtract accordingly.

For example, let us subtract 123 from 658. Both are three-digit numbers, therefore placing one number below the other in order of their place values, we get –

### Case 2 – The Minuend is a three-digit number while the subtrahend is either a one-digit number or a two-digit number

If the minuend is a three-digit number and the subtrahend is either a one-digit number or a two-digit number, we place one number below the other in order of their place values.

For example, let us subtract 21 from 794. Here the minuend, 794 is a three-digit number while the subtrahend, 21 is a two-digit number. So, in order to subtract a two-digit number from a three-digit number, we place the subtrahend below the minuend in the order of their place values and subtract accordingly.

Let us now subtract 4 from 237. Here the minuend, 237 is a three-digit number while the subtrahend, 4 is a one-digit number. So, in order to subtract a one-digit number from a three-digit number, we place the subtrahend below the minuend in the order of their place values and subtract accordingly.

## Subtraction of Numbers (With Borrowing)

Until now, we have discussed problems of subtractions where each digit of the subtrahend was less than the corresponding digit in the minuend. How would we subtract a subtrahend that is greater at a certain place value than the corresponding value of the minuend?

Let us understand this by an example. Suppose, we wish to subtract 2356 from 7814. Compare the corresponding digits of both the subtrahend and the minuend. You will notice that the value of subtrahend is larger than the value of the minuend at some places. So, how do we subtract 4 from 6? This is where borrowing comes into the picture. In such cases, we borrow 1 from the next number in the place value.

Let us solve this step by step.

1. Subtract 6 from 4. We cannot larger numbers, so we borrow 1 from the digit that is at the ten’s place in the minuend. In this case, the number is 1. So we borrow 1 from 1, and the 1 at the ten’s place in the minuend becomes 0. But the number at the one’s place of the minuend, after borrowing 1 becomes 14. Now we can subtract 6 from 14 and we get 8 at the one’s place as the answer.

2. Next, we move to subtract the digits at the ten’s place. Remember, the digit at the ten’s place is now 0, instead of the 1 that we had earlier. So, we need to subtract 5 from 0, which again is not possible. Therefore, we repeat the steps again that we did for subtracting the digits at the one’s place. We will borrow 1 from the digit at the hundred’s place, and give it to 0 at the ten’s place. So, 8 at the hundred’s place of the minuend becomes 7 and the 0 at the ten’s place of the minuend becomes 10. Now we can subtract 5 from 10 and we get 5 as the answer at the ten’s place.

3. Next, we look at the values at the hundred’s place. Here, we have 3 as the subtrahend and 7 as the minuend which can be easily subtracted. So, we get 4 as the answer at the hundred’s place.

4. Last, we check the values at the thousand’s place. We have 2 as the subtrahend and 7 as the minuend. Hence, we get 5 as the answer at the thousand’s place.

Hence, 7814 – 2356 = 5458

## Subtraction of Decimal Numbers

To subtract a decimal number, follow these steps.

1. Change to like decimals.
2. Line up the decimal points, that is the decimal numbers are placed one below the other such that the tens digit is below tens, ones is below ones, the decimal point is below the decimal point, the tenth digit is below the tenth digit, the hundredth digit is below the hundredth digit and so on.
3. Subtract as in case of the whole number, Borrow wherever necessary.
4. Place the decimal point in the difference directly below the decimal point in the decimal numbers.

For example, let us subtract 7.385 from 16.03

Step 1 – Converting to Like Decimals

We don’t make any changes to 7.385 but re-write 16.03 as 16.030

Step 2 – Line up the decimal points

We have,

Step 3 – Subtract the thousandths.

You cannot subtract 5 thousandths from 0 thousandths. Therefore, you need to borrow 1 from hundredths.

You have 3 at the hundredth’s place. Borrow 1 from 3 which now becomes 2. The 1 that you have borrowed from 3 goes to 0 at the thousandths place which now becomes 10.

Now you can subtract 5 from 10 and you get 5 as the difference at the thousandths place.

Step 4 – Subtract the hundredth’s place

You cannot subtract 8 hundredths from 2 at the hundredths place. Therefore, you need to borrow 1 from the tenth’s place.

But, you have 0 at the tenth’s place which means it has nothing to give it to you. So, you move on to the one’s place where you have 6.

Now, the borrowing of numbers will be completed in two steps.

1. First, from 6 at one’s place, you borrow 1 and give it to 0. The 6 at one’s place becomes 5 and the 0 at the tenths place becomes 10.
2. Now, that 0 at the tenth’s place has become 10, it can give 1 to the number 2 at the hundredth’s place. So, on borrowing 1 from 10 at tenth’s place, the number at tenth’s place becomes 9 and the number at hundredth’s place becomes 12.

Now you can subtract 8 from 12 and you get 4 as the difference at the hundredth’s place.

Step 5 – Subtract the tenth’s place.

After the last borrowing, the number at the tenth’s place is 9 which is greater than 3. Hence 3, upon being subtracted from 9 will be 6 as the difference at the tenth’s place.

Step 6 – Subtract the one’s place

You have 5 at the one place which is smaller than 7. Hence it is not possible to subtract 7 from 5. Now, check the digits at the ten’s place. It is 1. Borrow 1 from this digit. The digit at the ten’s place becomes 0, while the digit at the one’s place becomes 15 which is now greater than 7.

Now, subtract 7 from 15 and you get 8 as the difference at the one’s place.

Step 7 – Subtract the ten’s place.

The number at the ten’s place is 0 and there is nothing to subtract as well. So, you may leave it as it is or just write 0 as the difference.

Remember, the start of a decimal number has no value, so the answer to the difference of 16.030 and 7.385 becomes 8.645.

Remember

1. If you subtract 0 from a number the difference will be 0 itself. For example, if you subtract 0 from 8 you will get the answer as 8.
2. If you subtract 1 from a number, the difference would the number that precedes the given number. For example, if you subtract 1 from 52, the difference will be 51 which is the predecessor of 52.
3. Synonyms for the word subtract that are used in everyday life are “take away”, “decrease”, “minus”, “less” and “difference”.

## Properties of Subtraction

### Closure Property

Closure property states that when an operation is performed on two numbers, the resultant would also be of the same type as the numbers on whom the operation ash been performed.

This means that if you subtract one whole number from the other, the difference would also be a whole number. Let us check whether the closure property holds true for subtraction.

Consider two numbers, 5 and 7.

Subtract 5 from 7, we get the difference as 2 which is again a whole number.

Now subtract 7 from 5. Since 7 is larger than 5, we place the sign of minus (-) before the difference between the absolute values of the numbers.

Therefore,  5 – 7 = -2 which is not a whole number.

This means that closure property may or may not hold true for subtraction.

Hence we can say that subtraction does not satisfy the closure property.

### Commutative Property

Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter. This means that if one number is subtracted from the other, it does not matter which number is marked as the minuend and which one is the subtrahend. Does this hold true for subtraction?

Let us find out.

Suppose we have two numbers 32 and 15.

We want to subtract 15 from 32. The difference would be 32 – 15 = 17

Now, if we interchange the subtrahend and the minuend, we get the problem as 15 – 32.

The difference between 15 and 32 would be -17 which is not equal to 17.

This means that, if we have two numbers, a and b, then

a – b ≠ b – a

Hence, subtraction is not commutative for whole numbers.

### Associative property

Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter. This means that if we want to find the difference between three numbers, we will have to find the difference between any two first and then find the difference between the result of the first operation and the third number. Will we get the same answer for any order of three numbers a, b and c?

Let us find out.

Suppose, we have three numbers, 5, 8, and 9 and we want to find the difference between them.

We first perform the operation 5 – 8. We get the difference as -3. Now we find out the difference between this result (-3) and 9.

We have

-3 – 9 = -12 which is our final answer.

Now, let us move from right to left. Let us find out the difference between 9 and 8.

We get

9 – 8 = 1

Now we find the difference between 1 and the third number, 5

We have

1 – 5 = -4

Do we get the same answer as the numbers in the previous order? No. So what is does this mean?

This means that subtraction is not associative for whole numbers. The order of the numbers determines the result of the difference we intend to obtain between the numbers.