Write each of the following in the indicated base.
(a) 1021 three to base ten. ________________________
(b) 534 ten to base twelve __________________________
I will do one:
a) 1021b3 is 1+6+0+27 =34 base 10
the positions indicate the multiplier of that power of 3
1021=1*33+0*32+2*31 + 1*30
To convert a number from one base to another, you need to understand the concept of place values. Each digit in a number represents a different power of the base.
(a) Converting 1021 from base three to base ten:
In base three, the place values are powers of three:
3^3 | 3^2 | 3^1 | 3^0
27 | 9 | 3 | 1
To convert 1021 from base three to base ten, we multiply each digit by its corresponding place value and sum them up:
1 × 3^3 + 0 × 3^2 + 2 × 3^1 + 1 × 3^0 =
1 × 27 + 0 × 9 + 2 × 3 + 1 × 1 =
27 + 0 + 6 + 1 = 34
Therefore, 1021 in base three is equal to 34 in base ten.
(b) Converting 534 from base ten to base twelve:
In base twelve, the place values are powers of twelve:
12^2 | 12^1 | 12^0
144 | 12 | 1
To convert 534 from base ten to base twelve, we divide the number successively by powers of twelve, keeping track of the remainders:
534 ÷ 144 = 3 remainder 78
78 ÷ 12 = 6 remainder 6
6 ÷ 1 = 6 remainder 0
Now, read the remainders in reverse order. The remainders 6, 6, and 0 are the digits in the base twelve representation.
Therefore, 534 in base ten is equal to 660 in base twelve.