# Number bases

# 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**.