What is a complement?

The smallest number which, added to a given number, causes a carry into the next higher place. Computers use it to represent negative numbers.

How do I find the 9's and 10's complements?

9's complement: subtract each digit from 9. 10's complement: the 9's complement plus 1. For example the 9's complement of 635 is 364 and the 10's is 365.

How do I find the binary two's complement?

Flip every bit (0 to 1 and 1 to 0) to get the one's complement, then add 1. For example 0101 becomes 1010 then 1011.

Why does a computer subtract by adding?

It simplifies the electronic circuits: instead of a subtraction circuit it turns the number into its complement and only adds, which is faster and simpler.

Unit 6 · Lesson 5

Numeric operations (2)

5 steps. Each one is a short story and a question to answer.

Want to work out 10 − 7 without subtracting? There is a trick: the complement.

A complement is the smallest number that, when added to a number, causes a carry to the next place. Like 7 and 3 (7 + 3 = 10).

The 9's complement: subtract each digit from 9. For 635: 9−6=3, 9−3=6, 9−5=4, giving 364.

To check: 635 + 364 = 999. All nines, so it is right.

9's complement of 635

9's complement = subtract each digit from 9

What is the 9's complement of 635?

Pick an answer

The 9's complement is very close to what we want. One step is missing.

The 10's complement = the 9's complement + 1.

For 635: the 9's complement is 364, add 1 and the 10's complement is 365.

To check: 635 + 365 = 1000. A one followed by zeros, so it is right.

The number635

9's complement364

+ 1

10's complement365

10's complement = 9's complement + 1

What is the 10's complement of 635?

Pick an answer

The computer works in binary, so we need the same idea with zeros and ones.

The two's complement is the smallest number that causes a carry when added. It represents negative numbers and does subtraction.

First flip every bit (0 to 1, 1 to 0): that is the one's complement. 0101 becomes 1010.

Second add 1: 1010 + 1 = 1011. That is the two's complement.

Form

Value

Original

0101

One's complement (flip)

1010

Two's complement (+1)

1011

Two's complement = flip the bits (one's complement), then add 1

What is the two's complement of 0101?

Pick an answer

A computer doesn't really subtract. It only adds. How?

Steps: find the complement of the number being subtracted, then add it to the other number.

If a carry comes out of the top place, ignore it. What remains is the answer.

Binary example: 1000 − 0111. The complement of 0111 is 1001, add: 1000 + 1001 = 10001, drop the carry and get 0001.

1000 - 0111

Subtraction = add the complement, and drop the extra carry

If a carry appears beyond the top place after adding the complement, what do we do?

Pick an answer

Engineers want a simpler, faster computer. If subtraction had its own circuit, it would need an extra one.

The solution: the computer turns the subtrahend into its complement and then only adds.

So it doesn't need a subtraction circuit, and the electronics are simpler.

The two's complement is the most widely used way to represent negative numbers and subtract inside computers.

Without complements

An addition circuit

Plus a subtraction circuit

With two's complement

Just an addition circuit

Simpler and faster

Two's complement = negative numbers + subtraction by adding + simpler circuits

The main benefit of complements in a computer?

Pick an answer

Done, see the summary

Numeric operations (2)

Scroll down ↓ to continue

The lesson on one page

Every idea in the lesson with its rule, ready to screenshot and revise from.

1
The 9's complement
9's complement = subtract each digit from 9
The 10's complement
10's complement = 9's complement + 1
3
The binary two's complement
Two's complement = flip the bits (one's complement), then add 1
4
Subtracting by adding
Subtraction = add the complement, and drop the extra carry
Why computers use complements
Two's complement = negative numbers + subtraction by adding + simpler circuits

Test yourself

One question from every step. Answer and see your score.

Question 1 of 5

What is the 9's complement of 635?

Watch and listen

Quick reel

Unit 6

Learn what a complement is, the 9's and 10's complements, the binary two's complement, and how a computer subtracts by adding and represents negative numbers, in an animated explainer.

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The lesson in writing

The same explanation as the 3D scene, written out for quick review.

The 9's complement

9's complement = subtract each digit from 9. A complement is the smallest number that, when added to a number, causes a carry to the next place. Like 7 and 3 (7 + 3 = 10). The 9's complement: subtract each digit from 9. For 635: 9−6=3, 9−3=6, 9−5=4, giving 364.

Page 121

The 10's complement

10's complement = 9's complement + 1. The 10's complement = the 9's complement + 1. For 635: the 9's complement is 364, add 1 and the 10's complement is 365.

Page 122

The binary two's complement

Two's complement = flip the bits (one's complement), then add 1. The two's complement is the smallest number that causes a carry when added. It represents negative numbers and does subtraction. First flip every bit (0 to 1, 1 to 0): that is the one's complement. 0101 becomes 1010.

Page 123, 124

Subtracting by adding

Subtraction = add the complement, and drop the extra carry. Steps: find the complement of the number being subtracted, then add it to the other number. If a carry comes out of the top place, ignore it. What remains is the answer.

Page 125, 127

Why computers use complements

Two's complement = negative numbers + subtraction by adding + simpler circuits. The solution: the computer turns the subtrahend into its complement and then only adds. So it doesn't need a subtraction circuit, and the electronics are simpler.

Page 128

Questions about this lesson

What is a complement?: The smallest number which, added to a given number, causes a carry into the next higher place. Computers use it to represent negative numbers.
How do I find the 9's and 10's complements?: 9's complement: subtract each digit from 9. 10's complement: the 9's complement plus 1. For example the 9's complement of 635 is 364 and the 10's is 365.
How do I find the binary two's complement?: Flip every bit (0 to 1 and 1 to 0) to get the one's complement, then add 1. For example 0101 becomes 1010 then 1011.
Why does a computer subtract by adding?: It simplifies the electronic circuits: instead of a subtraction circuit it turns the number into its complement and only adds, which is faster and simpler.