Number System Conversion: Binary, Octal, Decimal and Hexadecimal

Every number system works the same way: each position is worth the base raised to a power, counting from 0 on the right. Decimal uses base 10, binary base 2, octal base 8 and hexadecimal base 16. Hexadecimal needs six extra symbols, so A to F stand for 10 to 15. Computer awareness sections in SSC, bank and railway exams test conversions between these bases, usually with small numbers that can be worked by hand in under a minute.

Learning this table removes most of the work: the binary-to-octal and binary-to-hex methods below are just lookups in it.

To convert binary to decimal, write the place values 1, 2, 4, 8, 16, 32, 64, 128 from the right and add the ones where the bit is 1.

To convert decimal to binary, divide by 2 repeatedly and write down each remainder. The binary number is the remainders read from the last to the first.

For small numbers, subtraction is quicker than division: take away the largest power of 2 that fits, mark that bit, and repeat with what is left.

Because 8 = 2³, each octal digit is exactly three bits; because 16 = 2⁴, each hex digit is exactly four bits. So conversions between binary and these bases need no arithmetic at all, only grouping. Always group from the right, adding zeros on the left if the last group is short.

As a cross-check, the original binary number's place values give 256 + 128 + 32 + 8 + 2 + 1 = 427 too.

To go between octal and hexadecimal, pass through binary: expand each octal digit to three bits, then regroup the bits in fours.

Binary addition follows four rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 10 (write 0, carry 1), and 1 + 1 + 1 = 11 (write 1, carry 1).

• ✘ Reading division remainders from top to bottom instead of bottom to top.
• ✘ Grouping bits from the left, which shifts every group when the length is not a multiple of 3 or 4.
• ✘ Treating hex letters as separate numbers, so that 3A becomes 310 instead of 58.
• ✘ Confusing the number of values n bits can hold (2ⁿ) with the largest value (2ⁿ − 1).

Converting the answer back to the original base takes a few seconds and catches nearly every slip. For the logic built on these bits, see logic gates and truth tables. For wider computer awareness revision, see computer and IT fundamentals and the computer awareness cheat code.

Why do computers use the binary number system?

Digital circuits work most reliably with two states, such as a voltage being present or absent. Binary needs only two digits, 0 and 1, so each bit maps directly onto one of those two states.

Why are octal and hexadecimal used if computers work in binary?

They are shorthand for binary. Because 8 = 2³ and 16 = 2⁴, every octal digit stands for exactly three bits and every hexadecimal digit for exactly four, so long binary strings become short and easier for people to read.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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