Suppose we have a number in base 10 and want to find out how to represent that number in, say, base 2.

How do we do this?

Well, there is a simple and easy method to follow. Let’s say I want to write 59 in base 2. My first step is to find the largest power of 2 that is less than 59.

So let’s go through the powers of 2:

1, 2, 4, 8, 16, 32, 64.

Okay, 64 is larger than 59 so we take one step back and get 32. 32 is the largest power of 2 that is still smaller than 59. How many “whole” (not partial or fractional) times can 32 go into 59?

It can go in only once because 2 x 32 = 64 which is larger than 59. So, we write down a 1.

1

Now, we subtract 32 from 59: 59 – (1)(32) = 27. And we move to the next lower power of 2. In this case, that would be 16. How many full times can 16 go into 27? Once. So we write down another 1 and repeat the process.

1

1

27 – (1)(16) = 11. The next lowest power of 2 is 8.

How many full times can 8 go into 11?

Once. So we write down another 1.

111

11

11 – (1)(8) = 3. The next lowest power of 2 is 4.

How many full times can 4 go into 3?

Zero.

So, we write down a 0.

1110

3 – (0)(4) = 3. The next lowest power of 2 is 2.

How many full times can 2 go into 3?

Once. So, we write down a 1.

11101

3 – (1)(2) = 1. And finally, the next lowest power of 2 is 1. How many full times can 1 go into 1?

Once. So, we write down a 1.

111011

1 – (1)(1) = 0. And now we stop since our next lowest power of 2 is a fraction.

This means we have fully written 59 in base 2.

Exercise

Now, try converting the following base 10 numbers into the required base

16 into base 4

16 into base 2

30 in base 4

49 in base 2

30 in base 3

44 in base 3

133 in base 5

100 in base 8

33 in base 2

19 in base 2

Solutions

1.100

2.10000

3.132

4.110001

5.1010

6.1122

7.1013

8.144

9.100001

10.10011