Numeral systems

Subject classification: this is an information technology resource .

Educational level: this is a secondary education resource.

Resource type: this resource is a lesson.

Completion status: this resource is ~75% complete.

This is a lesson in in the course Introduction to Computers, which is a part of The School of Computer Science

Introduction to Computers
What is a computer? Hardware Numeral systems Data sizes and speeds Data Coding and Information Decoding Types of computers Computer networks Software References

Objective

Familiarize the learner with method for expressing numbers and convert one method to another.

Introduction

The writing method for expressing numbers, is called the "Numeral System". In the first method we are writing numbers with combinations of 10 symbols {0,1,2,3,4,5,6,7,8,9},is called digits and numbers that are expressed with 10 digits, are called "decimal numeral or base-10 numbers".The other method are binary,Hexadecimal and ect.The binary numeral system, or base-2 number system, represents numeric values using two symbols: 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers.

Decimal Numeral System

In the first method discussed we write numbers with combinations of 10 symbols {0,1,2,3,4,5,6,7,8,9} called digits. Numbers that are expressed with 10 digits are called "base-10" numbers or "Decimal Numeral System". For example:

2 (one digit)

45 (two digit)

643 (three digit)

8785 (four digit)

etc.

In Decimal Numeral Systems, the value of a digit is multiplied according to its placement in a numerical sequence: (base-number ^ 0,1,2,3,...), from right to left.

First digit = (base-number ^ 0): 10^0 = 1

Second digit =(base-number ^ 1): 10^1 = 10

Third digit =(base-number ^ 2): 10^2 = 100

Fourth digit =(base-number ^ 3): 10^3 = 1000

etc.

For example:

20= (2*10)+(0*1)=20+0=20

456=(4*100)+(5*10)+(6*1)=400+50+6

84568=(8*10000)+(4*1000)+(5*100)+(6*10)+(8*1)=80000+4000+500+60+8

Binary Numeral System

Numbers expressed with 2 symbols (0, 1) are called binary, or "base-2" numbers.

For example:

1 (one-digit-read: 1)

10 (two-digit-read: 1, 0)

100 (three-digit-read: 1,0,0)

1101 (four-digit-read: 1, 1, 0, 1)

etc.

In the Binary Numeral System, digits have a value specified, this value being equal with (base-number ^ 0,1,2,3,...): (right to left)

First digit (base-number^0): 2^0 = 1

Second digit (base-number^1): 2^1 = 2

Third digit (base-number^2): 2^2 = 4

Fourth digit (base-number^3): 2^3 = 8

etc.

Converting Binary to Decimal

To convert binary to decimal, each digit is multiplied by the value of its position, and the results are added.

For example:

10 = (1*2)+(0*1)=2+0=2 → 10 (binary )= 2 (decimal)

101=(1*4)+(0*2)+(1*1)=4+0+1=5 → 101 (binary )= 5 (decimal )

11001=(1*16)+(1*8)+(0*4)+(0*2)+(1*1)=16+8+0+0+1=25 → 11001 (binary )= 25 (decimal)

111011=(1*32)+(1*16)+(1*8)+(0*4)+(1*2)+(1*1)=32+16+8+0+2+1=59 → 111011 (binary m)= 59 (decimal)

Converting Decimal to Binary

To convert decimal to binary

Divide the decimal number by 2

• If there IS a remainder the rightmost column will be a 1
  
• If there is NO remainder, the rightmost column will be a 0.
  

Then repeat the process, moving one column to the left each time until you have divided down to 1.

Example 1
15/2 = 7 remainder 1 (Binary number = ???1)
7/2 = 3 remainder 1 (Binary number = ??11)
3/2 = 1 remainder 1 (Binary number = ?111)
The final result will always be 1 in the leftmost column (Binary number = 1111)

Example 2 74/2 = 37 remainder 0 (Binary number = ??????0)
37/2 = 18 remainder 1 (Binary number = ?????10)
18/2 = 9 remainder 0 (Binary number = ????010)
9/2 = 4 remainder 1 (Binary number = ???1010)
4/2 = 2 remainder 0 (Binary number = ??01010)
2/2 = 1 remainder 0 (Binary number = ?001010)
The final result will always be 1 in the leftmost column (Binary number = 1001010)


NB - Although I've put ? in at each stage, you won't know how many columns are needed until you complete the process.

For a shortcut to see how many columns are needed, find the largest factor of 2 that is smaller than the decimal number you started with, e.g.
Example 1: The largest factor less than 74 is 64, which is 2 to the power 6. As the furthest right column is 2 to the power 0, this means we need 7 columns.

Hexadecimal Numeral System

Numbers written with 16 symbols {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} are called "base-16" numbers. For example:

A (one digit)

B5 (two digit)

6C3 (three digit)

AF85 (four digit)

etc.

so:
A(hexadecimal)=10(decimal).
B(hexadecimal)=11(decimal.
C(hexadecimal)=12(decimal).
D(hexadecimal)=13(decimal).
E(hexadecimal)=14(decimal).
F(hexadecimal)=15(decimal)

In the "Hexadecimal Numeral System", digits have a value specified, this value of digits are equal with (base-number^ 0,1,2,3,...):(right to left)

First digit (base-number ^ 0): 16^0 = 1

Second digit (base-number ^ 1): 16^1 = 16

Third digit (base-number ^ 2): 16^2 = 256

fourth digit (base-number ^ 3): 16^3 = 4096

etc.

Converting Hexadecimal to Decimal

To convert hexadecimal to decimal, each digit is multiplied by the value of its position, and the results are added.

For example:

A =10*1=10 → A(hexadecimal)=10(decimal)

B5=(11*16)+(5*1)=181 → B5(hexadecimal)=181(decimal)

6C3=(6*256)+(12*16)+(3*1)=1536+192+3=1731 → 6C3(hexadecimal)=1731(decimal)

AF85=(10*4096)+(15*256)+(8*16)+(5*1)=40960+3840+128+5=44933 → AF85(hexadecimal)=44933(decimal)

Converting Decimal to Hexadecimal

To convert decimal to hexadecimal, decimal number is divided by 16 and ...