write each of the following in the indicated base.
1011two to base ten
1574 to base twelve
MCMLIV to base ten
1011two to base ten
= 1(2^3) + 0(2^2) + 1(2) + 1
= 11
1574 to base twelve
powers of 12:
12, 144, 1728 , ...
1574/144 = 10 with remainder of 134
134/12 = 11 with remainder of 2
so 1574 = 10(12)^2 + 11(12) + 2
= AB2 in base 12
To convert numbers between different bases, follow these steps:
1. Determine the base of the given number. The base is usually indicated by the subscript after the number. For example, "1011two" means the number is in base 2 (binary), "1574" has no subscript, so it is assumed to be in base 10 (decimal), and "MCMLIV" is in base 10 (Roman numerals).
2. Use the place value system to convert the number. In each base, each digit carries a specific weight depending on its position.
Let's convert each of the given numbers to their indicated base:
1. 1011two to base ten (decimal):
To convert a binary (base 2) number to decimal (base 10), use the place value system.
In the binary system, the rightmost digit represents 2^0 (1), the second digit from the right represents 2^1 (2), the third digit represents 2^2 (4), and so on.
So, for 1011two:
1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 1 * 2^0 = 8 + 0 + 2 + 1 = 11
Therefore, 1011two is equivalent to 11 in base ten.
2. 1574 to base twelve:
Since the given number doesn't have a subscript, it is assumed to be in base ten (decimal).
To convert a decimal number to any other base, continuously divide the number by the desired base and collect the remainders. Start from the least significant digit and work your way up.
For example, to convert 1574 to base twelve:
Step 1: Divide 1574 by 12. The quotient is 131, and the remainder is 2. Write down the remainder.
Step 2: Divide 131 by 12. The quotient is 10, and the remainder is 11 (represented as 'A' in base twelve).
Step 3: Divide 10 by 12. The quotient is 0, and the remainder is 10 (represented as 'A' in base twelve).
Step 4: Since the quotient is now 0, we stop.
Reading the remainders from bottom to top, the base twelve representation of 1574 is "10A2".
3. MCMLIV to base ten (decimal):
MCMLIV is a Roman numeral, which represents numbers in base 10. However, the conversion from Roman numerals to decimal can be done by adding the respective values of the Roman symbols.
M = 1000
CM = 900
L = 50
IV = 4
So, adding up these values: 1000 + 900 + 50 + 4 = 1954
Therefore, MCMLIV is equivalent to 1954 in base ten.