Question

Convert the decimal number 147 base 10 to base 3. However, do this using a POWER SERIES EXPANSION of the form 147_10=a_2 *R^2+a_1*R^1+a_0*R^0, where R is the number 10 in the base of the number you want to convert to (3) (i.e., R=10_3) and {a_0, a_1,a_2} are the coefficients represented in base 3. Verify your answer using the "division method". Show all your work and all steps.

Answer 1

3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
147-81 is 66
66 < 81
so, 66-27 is 39
39 >= 27
so, 39-27 is 12
12 can be writen as 9+3
so, 147 = 1x81+2x27+1x9+1x3+0x1
= 1x3^4+2x3^3+1x3^2+1x3^1+0x3^0
so, in base-3 147 is 12110

Verification with division method
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Divide 147 successively by 3 until the quotient is 0
147/3 = 49, remainder is 0
49/3 = 16, remainder is 1
16/3 = 5, remainder is 1
5/3 = 1, remainder is 2
1/3 = 0, remainder is 1
Read remainders from the bottom to top as 12110

we can verify that it's correct!