Convert Between Arbitrary Bases

Convert Between Arbitrary Bases

When converting a number from one base to another, the key idea is to treat the digits as place values in the original base, then rebuild the value in the new base.

1) Understand the place values

For a number like (d_k d_{k-1} \dots d_1 d_0) in base (b), its value is [ d_k b^k + d_{k-1} b^{k-1} + \cdots + d_1 b + d_0. ] This works for any valid base, as long as each digit is smaller than the base.

2) Convert through a value you can work with

A reliable method is:

• Step A: Convert the original number to a standard value by expanding it using place values.
• Step B: Convert that value into the target base by repeated division, or by regrouping place values if the bases are related.

3) Keep digits valid

In the target base, each digit must be between (0) and (b-1). If a digit is too large, carry to the next place.

4) Check your answer

Convert your final answer back to the original base or to a standard value to confirm it matches.

Useful tip

Write every intermediate step clearly. This helps avoid digit mistakes, especially when bases use symbols for digits above 9.