# Number Bases

## Introduction to Number Bases

• A number base, or radix, is the number of unique digits, including zero, used to represent numbers in a positional numeral system.
• The decimal system, commonly used worldwide, is a base-10 system.
• Other common number bases include base-2 (binary), base-8 (octal), and base-16 (hexadecimal).

## Converting Between Bases

• To convert between bases, break the number into multiples of powers of the base and add them.
• Write each base-n number as the sum of base-n digits times powers of n.
• Use division and remainder calculations to transform a decimal into any base

## Base-n Arithmetic

• Arithmetic in any base largely follows the same rules as in base-10.
• Carryovers and borrowings occur whenever a digit is equal to or larger than the base.
• Practice with base-n addition, subtraction, multiplication, and division can build familiarity and fluency.

## Binary, Octal, and Hexadecimal Systems

• The binary system, base-2, uses only 0 and 1, and is fundamental to computer systems.
• The octal system, base-8, uses digits from 0 to 7. It was used in early computer systems and remains a convenient representation of binary data.
• The hexadecimal system, base-16, uses digits from 0 to 9 and letters from A to F. It’s often used in computing, for example, in web colours, coding, and digital displays.
• Familiarise with conversion between these bases, especially between binary, octal, and hexadecimal, because of their importance in computing.

## The Role of Zero

• Zero plays a key role in positional numeral systems. It allows the representation of numbers with no occurrences of some digit in the middle.
• Remember that handling zeroes can require special attention during arithmetic operations and conversions.

## Problems and Exercises

• Solve various problems and exercises involving number bases to reinforce understanding.
• Conceptual understanding is crucial with number bases; mechanical rule-following without understanding can easily lead to errors.

## Number Theory and Number Bases

• The study of number bases links deeply with number theory, and has important consequences in several areas.
• Understand the use of different number bases when examining divisibility, prime numbers, and modular arithmetic.