Bases – Upper primmary
Bases
We are accustomed to writing numbers in base ten, using the symbols for 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 probably because we have 10 fingers. For example, 75 means 7 tens and five units. However numbers can be written in any number base.
Example 1
Change 75 base ten to base eight
75ten = 113eight
| 8 | 75 | R |
| 8 | 9 | 3 |
| 1 | 1 |
Example 2
Change 113eight to base 10
113eight = [(1 x 82) + (1 x 81) + (3 x 80)] ten
= (1 x 64) + (1 x 8) +(3 x 1)
= 64 + 8 + 3
= 75
Therefore, if we use base 8 instead of base ten, then 75 is written as 113 which denotes one sixty-four (82 ), one eight (81 ) and 3 units (instead of hundreds, tens and units).
Base 2 is particularly useful as it only requires two symbols, for zero and one, and it is the way numbers are represented in computers.
Example 3
Change 75ten to base 2
| 2 | 75 | R |
| 2 | 37 | 1 |
| 2 | 18 | 1 |
| 2 | 9 | 0 |
| 2 | 4 | 1 |
| 2 | 2 | 0 |
| 1 | 0 |
Thus, 75ten = 1001011two
Just as, in base ten, the columns represent powers of 10 and have ‘place value’ 1, 10, 102 , 103 etc. (reading from right to left), so in base 2, the columns represent powers of 2. Hence the number 1001011 denotes (reading from right to left):
1 unit (20 ), 1 two (21 ), no fours (22 ), 1 eight (23 ), no sixteens (24 ), no thirty-twos (25 ), 1 sixty- four (26 ).
The number 1001011 in base 2 is the same as the number 75 in base ten.
Example 4
Change 75ten to base five
| 5 | 75 | R |
| 5 | 15 | 0 |
| 3 | 0 |
Thus, 75ten = 300five
We use the symbols 0, 1, 2, 3 and 4 to represent numbers in base 5. The columns in base 5 have ‘place value’ 1, 5, 25, 125, 625 etc. reading from right to left. The number 75 in base ten is the same as the number 300 in base five, that is 3 twenty-fives, no fives and no units.
Example 5
Change 203six to base ten
203six = [(2 x 62) + (0 x 61) + (3 x 60)]ten
= [2 x 36 + 0 + 3 x 1]ten
= 72 + 3 = 75ten
Writing the number 75 in base six we get 203, which represents 2 thirty-sixes, no sixes and 3 units.
We have seen that 75 (base10), 1001011 (base 2), 300 (base 5), 113 (base 8), and 203 (base 6) all represent the same number.
Similarly, we can write 75 in any base we choose and we can write all numbers in any base.
For revision questions and answers download PDF below
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