Number bases

Japanese school year: Math A

What you learn

You will learn how to represent numbers using different bases, such as binary and hexadecimal, and how to convert between them. This is fundamental in computer science and digital data representation. Prior familiarity with place value in the base-ten system and integer division is recommended.

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Key points

This formula converts a base- $n$ integer into base 10. Multiply each digit $a$ by its place value (a power of $n$ ) and sum them together.

(a_k a_{k-1} \dots a_0)_{(n)} = a_k n^k + a_{k-1} n^{k-1} + \dots + a_0

This rule defines the range of digits in base $n$ . Each digit $a_i$ must be an integer from 0 up to $n - 1$ (for example, base 2 uses only 0 and 1).

0 \le a_i < n

This formula converts a base- $n$ fraction into base 10. Divide each fractional digit $a$ by successive powers of $n$ (such as $n$ and $n^2$ ) and add them up.

(0.a_1 a_2 \dots a_m)_{(n)} = \frac{a_1}{n} + \frac{a_2}{n^2} + \dots + \frac{a_m}{n^m}

This is the procedure to convert a base-10 integer into base $n$ . Repeatedly divide the number by $n$ until the quotient is 0, then write down the remainders in reverse order.

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Practice 1Coming soon