Number bases study guide

A systematic method

A number should be interpreted together with its base. The same written symbols can have different values in different bases, so retain the subscript while working. In positional notation, each digit is multiplied by the value of its position and the results are added.

For a decimal-to-binary conversion, list powers of 2 from the largest needed value down to 2⁰. Place a 1 when that power can be subtracted from the remaining value and a 0 when it cannot. For 26₁₀, the relevant place values are 16, 8, 4, 2 and 1. The value uses 16, 8 and 2, so the result is 11010₂. Check it by calculating 16 + 8 + 2 = 26.

For a binary-to-decimal conversion, multiply each bit by its place value and add. For 11010₂, the contributions are 1×16 + 1×8 + 0×4 + 1×2 + 0×1, giving 26₁₀. Performing the reverse calculation is a useful check because both directions should agree.

Converting through hexadecimal

The most direct binary-to-hexadecimal method is to split the binary representation into groups of four from the right. If necessary, add zeroes to the left of the first group; these leading zeroes do not change the value. For 111001₂, write 0011 1001, then convert 0011 to 3 and 1001 to 9. The result is 39₁₆.

For hexadecimal-to-binary conversion, replace every hexadecimal digit with exactly four bits. For 3C₁₆, 3 becomes 0011 and C becomes 1100, producing 00111100₂. The leading zeroes can be important when showing the full four-bit group for each hexadecimal digit.

A hexadecimal digit has 16 possible values, matching the number of patterns available from four binary bits. This makes hexadecimal a shorthand for binary: each digit can stand for one complete group of four bits. It is therefore more compact to write AE₁₆ than 10101110₂, while preserving a direct relationship between the representations.

Exam application and self-check

When answering a conversion question, show the place values or the four-bit groups rather than giving only the final answer. Label the bases so that it is clear which representation is being used. For decimal, use powers of 10; for binary, use powers of 2; for hexadecimal, use powers of 16. In hexadecimal, translate A–F before calculating.

Self-check with these questions: Are all digits valid for the stated base? Have the rightmost positions been assigned 2⁰ or 16⁰? If binary is grouped for hexadecimal, are the groups counted from the right and padded on the left? Can the answer be converted back to the original representation? Finally, explain in your own words why four binary bits correspond to one hexadecimal digit, using the fact that base 16 has 16 possible digit values.