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History of Numbers

Number Bases

In some alien classrooms, the following sums were seen on the board:

base 2 arithmeticbase 8 arithmetic

base 12 arithmeticbase 16 arithmetic

What do these sums mean? Can you express them in a more familiar form?

Our numbers are expressed in base 10 or decimals. This means that we have only 10 numerals - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 - then we start again at 10. Each time we reach the next power of 10, we start again. However, numbers can be written in any base. In this project, you will look at some of those used by mathematicians and in industry, convert numbers between them, and see how easy or difficult it is to do calculations in them.

Binary

Binary numbers are the basis of all modern computers (why is this?), and consist of just the numerals 0 and 1. Whenever we get to a power of 2, we start again. The table below shows the decimal numbers from 0 to 15 in binary form.

Decimal number

Binary number

Decimal number

Binary number

0

0

8

1000

1

1

9

1001

2

10

10

1010

3

11

11

1011

4

100

12

1100

5

101

13

1101

6

110

14

1110

7

111

15

1111

Can you see how all these binary numbers are formed? Practise changing decimal numbers to binary numbers and vice versa. What do you think happens with numbers less than 1 - both 'decimals' and fractions? Can you do arithmetic calculations (addition, subtraction, multiplication, division, forming powers) with binary numbers? Do you find this easier than decimal arithmetic or harder?

Other bases

Find out what other number bases are in use - you could try octal (base 8), hexadecimal ( or hex, base 16), duodecimal (base 12) or base 60. Look at how the numbers are formed - especially for bases greater than 10, and how calculations are done in the bases you consider. Why do people use these other bases rather than base 10? Which are easiest to do calculations in?

Do you think you could use a fraction or an irrational number as a base? Try creating numbers using a half or the square root of 2 as a base.

Could you create an Excel spreadsheet which would convert numbers from any base chosen to any other?

Useful web links

http://schoolscience.rice.edu/duker/robots/binarynumber.html
This website gives an introduction to binary numbers in the context of robots.

http://en.wikipedia.org/wiki/Binary_numeral_system
This website gives an introduction to other number bases.

http://www.computerhope.com/binhex.htm
This website covers binary, hexadecimal and octal number systems.

http://www.absoluteastronomy.com/encyclopedia/n/nu/numeral_system.htm
This website covers many different number bases.