NUMBER SYSTEM

Learning Outcomes:

After reading the article readers will be able to:

• Recognize base of a number system.
• Number system with base 2,5,8 and 10
• Understand:
• Binary number system or base 2
• Number system with base 5
• Octal number system or base 8
• Base 10 or decimal number system
• Conversion (تحویل) of binary system, base 5 and base 8 to decimal system and vice Versa.
• Addition, subtraction and Multiplication of base 2,5 and 8.

NUMBER SYSTEMS: (عددی نظامات)

We know that any number can be formed with the help of 10 digits i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8 9. These numbers are called numerals (علامات) and these numerals are known as 'Arabic numerals'.(عربی علامات)

Base of a Number System: 

The number of digits involved (استعمال) in a number system is called the base of that number system.

 If a number system (عددی نظام) involves only two digits 0, 1, then base (اساس) is 2. A number system, in which 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used, is a system with base 10 (اساس دس کا نظام). 

Similarly, a number system (عددی نظام ) in which five digits 0, 1, 2, 3 and 4 are used is a system with base 5.

 Number System with Base 2, 5, 8 and 10: 

(a) Number System with Base 2: 

A number system formed by two digits 0, 1 is called Binary system (اساس دو کا نظام) and its base is 2. This system is not used in everyday(روزمرہ) life apparently(بظاہر). But it is very important number system because it is used in all types of computers(کمپیوٹر). Because computer stores(جمع کرنا) information in the form of binary numbers so the binary system is of primary importance in the modern age(جدید دور) of computer. 

(b) Number System with Base 5: 

This number system(عددی نظام) involves digits 0, 1 2, 3 and 4. The largest digit (سب سے بڑا عدد) in base 5 system is 4.

(c) Number System with Base 8: 

The number system(عددی نظام) with base 8 is called octal system. In this system(نظام) eight digits 0, 1, 2, 3, 4, 5, 6 and 7 are used. The largest (سب سے بڑا) digit in octal system is 7.

(d) Decimal Number System:

 Decimal number system (اعشاری نظام ) is the most popular number system in the world. In this system (نظام) ten digits (0 to 9) are used. Every number can be expressed (ظاہر کرنا) as the sum of multiples (اضعاف) of powers of 10 and 10 is called its base. 

CONVERSIONS: (تحویلات)

The above discussed (بیان کیا گیا) number systems are all place value number systems. The numbers used in these systems can be converted (تبدیل کرنا) from one system to another system. The method of successive division (مسلسل تقسیم) is used to convert a number from one system to another system. The division (تقسیم) is performed by the base of the system in which it is being converted.

Conversion from decimal number system (اعشاری نظام) to other number systems:

A) Conversion from decimal (base 10) to binary system:

Example: Convert 15 into binary system or base 2.

Solution:

We write the numbers from bottom to top.

Example: change 541 into base 2 or binary system.

Solution:

Thus 541 = (1000011101)₂

B) Conversion from decimal system (base 10) to a number with base 5:

Any number of decimal system can be converted to equivalent no. with base 5 as

Example: Convert 17 to an equivalent no. with base 5

Solution:

Example: Convert 89651 to an equivalent no. with base 5

Solution:

Thus 89651 = ( 10332101)₅

C) Conversion from decimal to base 8 or Octal system:

Example: convert 824 to an equivalent no. with base 8(Octal system)

Solution:

Hence 824 = (1470)₈

Example: Convert 4837 to an equivalent number with base 8 (Octal system).

Solution:

Result, 4837 = (11345)₈

Conversion (تحویل) from other Number system to decimal (اعشاری) number system:

A) Conversion from binary system to decimal system:

We can understand this conversion by following examples

Example: change (1101)₂ into equivalent number in decimal system.

Solution:

(1101)₂ =( 1×2³) + (1×2²)+(0×2¹)+(1×2⁰)

(1101)₂ = 8+4+0+1 = 13

B) From base 5 to decimal system (base 10)

Example: change (413242)₅ into equivalent decimal system.

Solution: (413242)₅ = (4×5⁵)+(1×5⁴)+(3×5³)+(2×5²)+(4×5¹)+(2×5⁰)

(413242)₅ = (4×3125)+(1×625)+(3×125)+(2×25)+(4×5)+(2×1)

(413242)₅ = 12500+625+375+50+20+2 = 13572

C) Conversion from Octal system (base 8) to decimal system:

Example: convert (126)₈ to decimal system.

Solution: (126)₈ = 1×8²+2×8¹+6×8⁰

(126)₈ = (1×64)+(2×8)+(6×1)

(126)₈ = 64+16+6 = 86

Example: Convert (424002)₈ to decimal system.

Solution: (424002)₈ = (4×8⁵)+(2×8⁴)+(4×8³)+(0×8²)+(0×8¹)+(2×8⁰)

(424002)₈ = (4×32768)+(2×4096)+(4×512)+(0)+(0)+(2×1)

(424002)₈ = 131072+8192+2048+0+0+2 = 141314

Addition, Subtraction and Multiplication:

A) Addition base 2:

There are only two digits 0 and 1 used in the binary number system.

When we add the numbers and if the sum is greater than 1 then divide the sum by 2, write the remainder and carry quotient to next digit.

Following examples are helpful to understand the addition process of binary number system.

Example: add these two numbers of binary system. (111)₂ and (10)₂

Solution:

Example: solve: (10110111)₂ + (100011)₂

Solution:

B) Subtraction of binary number system:

Example: solve (101)₂ - (11)₂

Solution:

Example: find (10011)₂ - (1101)₂

Solution:

C) Multiplication of base 2:

Example: solve (11)₂ × (10)₂

Solution:

Example: multiply (11011011)₂ by (10101)₂

Solution:

Base 5 (Octal system) number system:

A) Addition of base 5

In addition if the sum of two or more digits is greater than 5 then divide the sum by 5 and write the remainder and carry the quotient to the next digit.

Example: Find sum of (4)₅ and (3)₅

Solution:

Example: solve (12433)₅ + (31243)₅

Solution:

B) Subtraction of base 5:

Example: solve (3421)₅ - (2143)₅

Solution:

C) Multiplication of base 5:

Example: solve (23)₅ × (14)₅

Solution:

Example:

Multiply (421)₅ by (234)₅

Solution:

Base 8 OR Octal number system:

In Octal system there ae eight Numbers 0,1,2,3,4,5,6,7.

A) Addition in base 8 OR Octal system:

Example: Solve (6)₈ + (7)₈

Solution:

Example: (64)₈ + (44)₈

Solution:

Example: Solve (255636)₈ + (143576)₈

Solution:

B) Subtraction in base 8 OR Octal number system:

Example: Solve (14)₈ - (6)₈

Solution:

Example: (604)₈ - (247)₈

Solution:

Example: (455122)₈ - (216634)₈

Solution:

C) Multiplication in base 8 OR Octal number system:

Example: solve (36)₈ × (43)₈

Solution:

Example: (446)₈ × (213)₈

Solution:

Addition(جمع), Subtraction (تفریق) and Multiplication (ضرب) of numbers with different bases:

For this first of all we convert all the numbers into decimal system OR base 10 and perform the given operations. Then the answer is converted into base 2, base 5 and base 8 as required.

Example: solve (100111)₂ + (4123)₅ + 567 and give answers in base 2 ,base 5 and base 10.

Solution: We convert (100111)₂ and (4123)₅ into base 10 OR decimal system.

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example. solve (777)₈ - (2343)₅ - (1000111)₂ and give your answer in binary system (base 2).

Have a nice day.