Write each of the following in the indicated base.
(a) 1021three to base ten.
(b) 534 to base twelve
(c) MCMLXII to base ten
a) 10213 = 1(3^3) + 0(3^2) +2(3) + 1
= 27+6+1
= 34
b) powers of 12 are 12, 144 , 1728 ,....
so 534 = 3(144) + 8(12) + 6
= 38612
c) MCMLXII = 1000 + 900 + 50 + 12 = 1962
To convert numbers between different bases, follow these steps:
(a) Convert 1021base three to base ten:
To convert a number from base three to base ten, we multiply each digit of the number by the corresponding power of three and sum them up.
1021base three = 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, 1021base three is equal to 34 in base ten.
(b) Convert 534 to base twelve:
To convert a number from base ten to base twelve, we divide the number successively by twelve and keep track of the remainders. The remainders will represent the digits in the new base.
534 / 12 = 44 with a remainder of 6 (6 is the least significant digit, as it's the remainder of the first division)
44 / 12 = 3 with a remainder of 8 (8 is the next digit, as it's the remainder of the second division)
3 / 12 = 0 with a remainder of 3 (3 is the most significant digit, as it's the remainder of the final division)
Therefore, 534 in base ten is equal to 383 in base twelve.
(c) Convert MCMLXII to base ten:
To convert a number from a non-decimal base to base ten, we need to convert each digit of the number to their corresponding numeric value in base ten and add them up.
M = 1000
C = 100
M = 1000
L = 50
X = 10
I = 1
I = 1
MCMLXII = 1000 + 100 + 1000 + 50 + 10 + 1 + 1
= 3162
Therefore, MCMLXII in base ten is equal to 3162.