Unit 1 Section 1 : Binary Numbers

We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers.

In base 10 we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

In base 2 we use only the digits 0 and 1.

Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents no current flowing whereas 1 represents a current flowing.

In base 10 we use a system of place values as shown below:

1000	100	10	1
4	2	1	5		4 × 1000 + 2 × 100 + 1 × 10 + 5 × 1
3	1	0	2		3 × 1000 + 1 × 100 + 2 × 1

Note that, to obtain the place value for the next digit to the left, we multiply by 10.
If we were to add another digit to the front (left) of the numbers above, that number would represent 10 000s.

In base 2 we use a system of place values as shown below:

64	32	16	8	4	2	1
1	0	0	0	0	0	0		1 × 64 = 64
1	0	0	1	0	0	1		1 × 64 + 1 × 8 + 1 × 1 = 73

Note that the place values begin with 1 and are multiplied by 2 as you move to the left.

Once you know how the place value system works, you can convert binary numbers to base 10, and vice versa.

Example 1

Convert the following binary numbers to base 10:

(a)	111
(b)	101
(c)	1100110

Example 2

Convert the following base 10 numbers into binary numbers:

(a)	3
(b)	11
(c)	140

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.