NUMBER BASES

The decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.
[1] CONVERSION FROM OTHER BASE LESS THAN TEN TO BASE TEN
Firstly, we shall consider conversion of numbers in a base less than ten to a base ten number. A number in base ten is known as a decimal or denary number. All numbers in a given n can be written using only the  following digits 0, 1, 2 , ….., n -1. For instance in base two, the only digits that can be used are only 0 and 1. In base three, you can only use digits 0, 1 or 2.

Generally our normal counting is done in base ten when doing this, the base is normally indicated.

E.g in the denary number 546 is 546ten.  The first digit from the right towards left is 6 and is called the unit digit. The next digit is 4 and is called the tens digit and has value 4x10.

The next digit 5 is called the hundred digit i.e 500. Hence 52 41 60ten= 5 4 6 = 5x102+4x101+6x100
= 500 + 400+ 6
= 546
Notice that the digits listed on top of the denary numbers 546 are the powers of the base.

SAMPLE 1
Convert 1203 in base five to a denary number.
Solution
Circle the digits of the number 1203 from the last to the first, beginning at zero
i.e203five
Therefore expand number raising the base five to the grades listed on the top of the number as shown below;
2 0 3 five= 1x53+2x52+0x51+3x50
= 1x125+2x25+0x5+3x1
= 125+50+0+3
= 178
The new number is in base ten i.e 1203five=  178ten   
This expansion method can be used in converting from any base to base ten.

EXERCISE 1
Convert the following to denary numbers
1. 10.2three
2. 214seven
3. 780nine
4.Find if 200x + 144nine = 14Btwelve.
5.Solve for x and y if 32x + 53y + 61nine
24x + 35y = 45ten

[2] CONVERSION FROM OTHER BASE GREATER THAN TEN TO BASE  TEN
Expansion method can be used to convert numbers in base say base thirteen to base ten, Remember in base thirteen kthe digits we have are 0,  1, 2, 3,4,5, 6,7,8,9, A, B, C. where A represents ten B represents eleven and C represents twelve. Letters are used for two- digits numbers less than the base thirteen.

SAMPLE 2
Convert 1B9thirteen to denary number
Solution
1B9thirteen= 1x132+Bx131+9x130
= 1x169+11x13+9x1
= 169+143+9
= 321ten

SAMPLE 3
Convert 20Cfifteen to a denary number
Solution
20Cfifteen= 2x152+0x151+12x150
= 2x225+0x15+12x1
= 450+0x15+12x1
= 462ten

EXERCISE 2
1. Convert the following to denary numbers
[I] 1024eleven
[II] 2059twelve
[III] 51Cfourteen

2. Convert the following numbers to denary numbers
[I] 10011two
[II] 768nine
[III] 10Aeleven
[IV] B12twelve
[V] 7B3Atwelve
[VI] 6D4Fsixteen

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