[PDF] 3 Convert a number from one number system to another PDF Lecture_4_Digital_Number_Systems

for example: we use 10-based numbering system for input and output in digital calculator There are two ways to convert a decimal number to its equivalent quotient of 0 is obtained Note: The binary result is obtained by writing the first

A) Divide the Number (Decimal Number) by the base of target base system (in which you want to convert the number: Binary (2), octal (8) and Hexadecimal (16 ))

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NUMBER SYSTEM CONVERSION There are many methods or techniques which can be used to convert numbers from one base to another We'll demonstrate

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Uses 16 symbols: 0, 1, 2, , 9, A, B, C, D, E, F Notes: ▫ When written down, a number may be ambiguous regarding which system it belongs to So we will

[PDF] 3 Convert a number from one number system to another

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number system, e g , it may be required to convert a decimal number to binary or octal or hexadecimal The reverse is also true, i e , a binary number may be

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The base or radix of number system determines how many numerical digits the number To convert a decimal number to any other number system, divide the decimal And note down the remainders in the reverse order • To convert from

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Replace each hexadecimal digit with the corresponding 4-bit binary string 8B16 = 1000 1011 = 100010112 Page 2 Conversion of Decimal Numbers

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Moreover, we can always convert them from any high-base number system to a Note that since these number systems possess base 2k , all numbers within

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Binary : 7 3 1 Octal : 111011 001 So, 731, = 1110110012 5 Decimal to Octal Conversion: To convert a decimal number to its octal equivalent, the remainder

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3. Convert a number from one number system to another

Conversion between number bases:

Way we need conversion?

9 We need decimal system for real world (for presentation and input):

for example: we use 10-based numbering system for input and output in digital calculator.

9 We need binary system inside calculator for calculation.

Decimal

(10)

Binary (2)

Octal (8)

Hexa (16)

More Interest

Input Conversion

10-to-2

Conversion

2-to-10

Digital

System

Output

a) Binary to decimal conversions:

9 Rule: any binary number can be converted to its decimal equivalent

simply by summing together the weights of the various positions in the binary number which contains a 1. Example 1: Convert 110112 to its decimal equivalent

1 1 0 1 1

+ + + = 16+8+2+1= 2710

24 +23 0 21 20

Example 2: Convert 101101012 to decimal equivalent

27 + 0+ 25+ 24+ 0+ 22+ 0+ 20 = 18110

b) Decimal to binary conversions:

9 There are two ways to convert a decimal number to its equivalent

binary representation

1. The reverse of the binary-to-decimal conversion process

(optional). The decimal number is simply expressed as a sum of powers of 2 and then 12 and 02 are written in the appropriate bit positions.

Example 1:-Convert 4510 to binary number

4510 = 32+ 8+ 4+ 1= 25+ 0+ 23+ 22+ 0+ 20 =101101(2)

Example 2:-Convert 7610 to binary number

7610= 64+ 8+ 4 = 26+ 23+ 22 = 10011002

2. Repeated division: Repeating division the decimal number by

2 and writing down the remainder after each division until a

quotient of 0 is obtained. Note: The binary result is obtained by writing the first remainder as the LSB and the last remainder as the MSB. General Rule 1: Conversion from decimal to other base

1. GLYLGH GHŃLPMO QXPNHU N\ POH NMVH 2 8 16"B

2. The remainder is the lowest-order digit.

3. Repeat first two steps unit no divisor remains.

General Rule 2: Decimal fraction conversion to anther base

1. 0XOPLSO\ GHŃLPMO QXPNHU N\ POH NMVH 2 8".

2. The integer is the highest-order digit.

3. Repeat first two steps until fraction becomes

zero.

Example 1 Convert 2510 to binary number

2510 = 1 1 0 0 12

Example 2 Convert 1310 to binary number

Division by 2 Quotient integer remainder

૛ = 6 1 (a0) ૛ 3 0 (a1) ૛ 1 1 (a2) ૛ 0 1 (a3)

Answer (13)10 = (a3 a2 a1 a0) = (1101)2

Example 3: Convert 0.62510 to binary number

Multiply by 2 Integer Fraction coefficient

0.625*2 = 1 + 0.25 a1 = 1

0.250*2 = 0 + 0.50 a2 = 0

0.500*2 = 1 + 0(stop) a3 = 1

Answer (0.625)10 = (0.a1 a2 a3)2 = (0.101)2

c) Octal-to-decimal

9 To convert, we need to multiply each octal digit by its positional

weight.

Example 1

372(8) = (382) + (781) + (280) = (364) + 56+2 = 25010

Example 2

24.68 = (281) + (480) + (6*8-1) = 20.7510

d) Decimal to octal

9 Repeated division by 8.

Example 1: Convert 26610 to octal number.

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[PDF] [PDF] 3 Convert a number from one number system to another