NUMBER NUMBER SYSTEMSYSTEMNUMBER NUMBER SYSTEMSYSTEM

The mysterious world of numbers…

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AcknowledgementWe would like to thank AKP sir for giving us an opportunity to express ourselves on this enthusiastic project. Any accomplishment requires the effort of many people and this work is no different. Every group member has been an important part of this project. We also thank our friends for their ideas and co-operation they provided to us. We are grateful to all of them.

Thank you..

A number is a mathematical object used in counting and measuring. Numerals are often used for labels, for ordering serial numbers, and for codes like ISBNs.

In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

The number system we use on day-to-day basis in the decimal system , which is based on ten digits: zero through nine. As the decimal system is based on ten digits, it is said to be base -10 or radix-10. Outside of specialized requirement such as computing , base-10 numbering system have been adopted almost universally. The decimal system with which we are fated is a place-value system, which means that the value of a particular digit depends both on the itself and on its position within the number.

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. Owing to its straight forward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers.

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1.

Archimedes :

He was a Greek mathematician. He was the first to compute the digits in the decimal expansion of π (pi). He showed that -

3.140845 < π < 3.142857

Archimed

es

Conversion Among Bases

•The possibilities:

Hexadecimal

Decimal Octal

Binary

Quick Example

2510 = 110012 = 318 =1916

Base

Different conversions Different conversions possible:possible:

> Binary to decimal> Octal to decimal> Hexadecimal to decimal> Decimal to binary> Octal to binary> Hexadecimal to binary> Decimal to octal, etc..

Technique◦ Multiply each bit by 2n, where n is the “weight” of

the bit◦ The weight is the position of the bit, starting from

0 on the right◦ Add the results

1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32

4310

Bit “0”

Octal to DecimalOctal to DecimalTechnique

◦Multiply each bit by 8n, where n is the “weight” of the bit

◦The weight is the position of the bit, starting from 0 on the right

◦Add the results

7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448

46810

TechniqueMultiply each bit by 16n, where n is the

“weight” of the bitThe weight is the position of the bit, starting

from 0 on the rightAdd the results

ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560

274810

TechniqueDivide by two, keep track of the remainderFirst remainder is bit 0 (LSB, least-significant

bit)Second remainder is bit 1Etc.

12510 = ?2

2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1

12510 = 11111012

Fractions

Decimal to binary3.14579

.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056

etc.11.001001...

Octal to BinaryOctal to BinaryTechnique

◦Convert each octal digit to a 3-bit equivalent binary representation

ExampleExample7058 = ?2

7 0 5

111 000 101

7058 = 1110001012

TechniqueConvert each hexadecimal digit to a 4-bit

equivalent binary representation

10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112

Common Powers (2 of 2)

Base 2Power Preface Symbol

210 kilo k

220 mega M

230 Giga G

Value

1024

1048576

1073741824

What is the value of “k”, “M”, and “G”?

In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

Example

/ 230 =

In the lab…1. Double click on My Computer2. Right click on C:3. Click on Properties

Numbers are never ending. You look into it, you find a world of quantities, helping you in your daily chores.

It’s a simple yet hard to understand, you work on it, you are going to love it more and more.

Exploring it is the best option, so just enjoy it.

Project made and Compiled by ~

Sanjana PoddarSana Jahan

Ronodeep MazumdarRiya Debnath