Some familiar number systems are radix 10 (decimal), radix 16 (hexadecimal), radix 8 (octal), and radix 2 (binary). Numbers are represented by a string of digit symbols in a given radix. Given a radix r, there are r different symbols available. For a string of digits of length n, there are rⁿ different numbers that can be represented. The following table shows various representations of the first 16 numbers:

We can easily convert from binary to octal and hexadecimal using three and four digit lookups. Consider the binary number 1010 1111 0111 0010. A three digit lookup gives us the octal representation:

A four digit lookup gives us the hexadecimal representation:

We can check our answers by converting each to decimal:

A*16³ + F*16² + 7*16¹ + 2*16⁰ = 44914

1*8⁵ + 2*8⁴ + 7*8³ + 5*8² + 6*8¹ + 2*8⁰ = 44914

2¹⁵ + 2¹³ + 2¹¹ + 2¹⁰ + 2⁹ + 2⁸ + 2⁶ + 2⁵ + 2⁴ + 2¹ = 44914

To convert from decimal to a given base, use repeated integer division by the base of conversion recording the remainder each step until the quotient is 0 and you have the final remainder. The first remainder out is adjacent to the radix point (the low order digit).

Complements in Radix r	Euclid's Division Algorithm	Base Conversion	Practice in two's complement
Java Primitive Data Types:	Integers	Reals	Homework

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by J. A. Tompkins tompkinsj@uncw.edu