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Number Systems

Number systems are a mathematical system for expressing numbers. A number system consists of a set of symbols that are used to represent numbers, and a set of rules for manipulating those symbols. The symbols used in a number system are called numerals.

Table of Contents

Types of Number Systems

Positional Numeral System

Positional Notation most commonly uses the decinal system. The symbols used in the decimal number system are the Hindu-Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is positional numeral system and is base 10, which means that each place in the number represents a power of 10, also known as the radix. The place value of each digit is determined by its position in the number, with the most significant digit on the left and the least significant digit on the right. The value of a digit is multiplied by the radix raised to the power of its position. For example, in the number 123, the digit 1 is in the hundreds place, the digit 2 is in the tens place, and the digit 3 is in the ones place. The value of the digit 1 is 100, the value of the digit 2 is 20, and the value of the digit 3 is 3. The sum of these values is 123.

Sign-value Notation System

Sign-value notation represents numbers by a series of numeric signs that when added together give the value of the number. The most commonly known exmaple would be Roman Numerals, where I means 1, V means 5, X means 10. Thus IV means 4, IX means 9, and XI means 11. Sign-value notation is mostly historical and is not used commonly in the modern world.

Common Positional Number Systems

Binary

Binary Numbers only have two v, 0 and 1. Binary numbers are used in computers because they are easy to represent using electronic circuits. Binary numbers are also used in digital electronics, such as digital cameras, digital watches, and digital televisions.

Octal

Octal Numbers have eight characters, 0, 1, 2, 3, 4, 5, 6, and 7. The main advantage of using octal numbers is that they are easier to work with than binary numbers. Octal numbers are used in computer programming and in the Unix operating system.

Decimal

Decimal Numbers have ten characters, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Decimal numbers are commonly used in everyday life.

Hexadecimal

Hexadecimal Numbers have sixteen characters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The Hexadecimal, or Hex, numbering system is commonly used in computer and digital systems to reduce large strings of binary numbers into a sets of four digits for us to easily read.

Deciaml Binary Octal Hexadecimal
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

Base/Radix Conversion

Base/Radix conversion is the process to convert the number from one base to another base. For example, converting a decimal number to binary number. These are some example videos on the common types.

Youtube Video Playlist

Radix to Decimal

The basic formula for another radix to decimal conversion is:
$a = r_m b^m + r_{m-1}b^{m-1} + r_{m-2}b^{m-2} + r_{1}b^{1} + r_{0}b^{0}$
where, m is a nonnegative integer and the r's are integers such that
$0 < r_m < b$ and $0 ≤ r_i < b for i = 0, 1, ... , m − 1$.

Example: Convert 2212 from base 3 (radix_3) to decimal (radix_10)

$a = (2)3^3+(2)3^2+(1)3^1+(2)3^0$
$a = 77$

Thus $2212_{3} = 77_{10}$.

Decimal to Radix

The basic algorithm for decimal to another radix conversion is to take the decimal number and divide it by the new base/radix. The remainder is the rightmost digit of the new number. The quotient is the new number to be converted. The process is repeated until the quotient is zero.

Example: Convert 77 from decimal (radix_10) to base 5 (radix_5)

$77/5 = 15b + 2$   the remainder is 2, so the rightmost digit of the new number is 2. The quotient is 15, so the new number to be converted is 15.
$a = 2$
$15/5 = 3b + 0$    the remainder is 0, so the rightmost digit of the new number is 0. The quotient is 3, so the new number to be converted is 3.
$a = 02$
$3/5 = 0b + 3$      the remainder is 3, so the rightmost digit of the new number is 3. The quotient is 0, so the new number to be converted is 0.
$a = 302$

Thus $77_{10} = 302_{5}$.

Mixed-Base Number Systems

Mixed-base number systems are number systems that use more than one base. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. The easiet example of a mixed-base number system is from our timekeeping methods. 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, 7 days in a week, 4 weeks in a month, 12 months in a year, 10 years in a decade, 10 decades in a century, 10 centuries in a millennium. All of these are different bases and they are all combined to make our timekeeping system.