How it works
In base b, the place values from the right are 1, b, b², b³, … and the digits run from 0 to b − 1. So 123 in base 7 means 1 × 49 + 2 × 7 + 3 = 66 in base 10.
To base 10: multiply each digit by its place value and add. From base 10: either subtract the largest place values that fit (13 = 8 + 4 + 1, so 13 = 1101 in binary), or divide by the base repeatedly and read the remainders from bottom to top.
Binary (base 2) is the most common contest base. It is worth knowing the powers of 2 up to 1024 by heart.
Number Bases practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
What is base 7 number 123 in base 10?
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Answer: 66
1 × 7² + 2 × 7 + 3 = 49 + 14 + 3 = 66.
What is base 10 number 13 in binary, written as a decimal number using only digits 0 and 1?
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Answer: 1101
13 = 8 + 4 + 1 = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1, so it is 1101 in binary.
What is base 5 number 4321 in base 10?
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Answer: 586
4 × 125 + 3 × 25 + 2 × 5 + 1 = 500 + 75 + 10 + 1 = 586.
What is binary 1101 plus binary 1011, written in base 10?
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Answer: 24
1101₂ = 8 + 4 + 1 = 13 and 1011₂ = 8 + 2 + 1 = 11. The sum is 24.
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