Example: 243_{(5)} = 2 x 5^{2} + 4 x 5^{1} + 3 x 5^{0} = 2 x 25 + 4 x 5 + 3 x 1 = 50 + 20 + 3 = 73
1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 1 + 4 = 10 2 + 1 = 3 2 + 2 = 4 2 + 3 = 10 2 + 4 = 11 3 + 1 = 4 3 + 2 = 10 3 + 3 = 11 3 + 4 = 12 4 + 1 = 10 4 + 2 = 11 4 + 3 = 12 4 + 4 = 13
12 = 1 x 5^{1} + 2 x 5^{0} = 7_{(10)} + 2 = 2 x 5^{0} = 2_{(10)} ------- 14 = 1 x 5^{1} + 4 x 5^{0} = 9_{(10)}
12 = 1 x 5^{1} + 2 x 5^{0} = 7_{(10)} + 3 = 3 x 5^{0} = 3_{(10)} ------- 20 = 2 x 5^{1} + 0 x 5^{0} = 10_{(10)} How to add: 2 + 3 = 10, write down 0 and remember carry 1 1 + carry 1 = 2
141 = 1 x 5^{2} + 4 x 5^{1} + 1 x 5^{0} = 46_{(10)} + 2340 = 2 x 5^{3} + 3 x 5^{2} + 4 x 5^{1} + 0 x 5^{0} = 345_{(10)} ------- 3031 = 3 x 5^{3} + 0 x 5^{2} + 3 x 5^{1} + 1 x 5^{0} = 391_{(10)} How to add: 1 + 0 = 1 4 + 4 = 13, write down 3, remember carry 1 1 + 3 = 4, add carry 1 = 10, write down 0, carry 1 2 + carry 1 = 3
| 1 2 3 4 ----+------------------- 1 | 1 2 3 4 2 | 2 4 11 13 3 | 3 11 14 22 4 | 4 13 22 31
2 x 3 = value 6 ----- which is represented in base-5 as 11 (1 x 5 + 1 = 6)
2 x 4 = value 8 ----- which is represented in base-5 as 13 (1 x 5 + 3 = 8)
3 x 3 = value 9 ----- which is represented in base-5 as 14 (1 x 5 + 4 = 9)
3 x 4 = value 12 ----- which is represented in base-5 as 22 (2 x 5 + 2 = 12)
4 x 4 = value 16 ----- which is represented in base-5 as 31 (3 x 5 + 1 = 16)
Example 1:
Base-5 Value 23 2 x 5 + 3 = 13_{(10)} x 4 = 4_{(10)} ---- 202 4 x 3 = 22, write down 2 and remember carry 2 4 x 2 = 13, add the carry 2: 13 + 2 = 20 write down 0 and remember carry 2In other words: 23_{(5)} x 4_{(5)} = 202_{(5)}
We can check the correctness:
202_{(5)} = 2 x 5^{2} + 0 x 5^{1} + 2 x 5^{0} = 2 x 25 + 2 = 52_{(10)} (which is equal to 13_{(10)} x 4_{(10)})
Example 2:
Base-5 Value 43 4 x 5 + 3 = 23_{(10)} x 32 3 x 5 + 2 = 17_{(10)} ---- 141 2340 ----- 3031In other words: 43_{(5)} x 32_{(5)} = 3031_{(5)}
We can check the correctness:
3021_{(5)} = 3 x 5^{3} + 0 x 5^{2} + 3 x 5^{1} + 1 x 5^{0} = 3 x 125 + 0 x 25 + 3 x 5 + 1 = 375 + 15 + 1 = 391_{(10)} (which is equal to 23_{(10)} x 17_{(10)})