Exploring Number Bases: Converting Decimal to Binary, Octal, and Hex, Study notes of Computer Science

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CSE1520 W08 M. Mandelbaum

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Chapter 2

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Chapter Goals „^ Know the different types of numbers „^ Describe positional notation „^ Convert numbers in other bases to base 10 „^ Convert base 10 numbers into numbers of otherbases „^ Describe the relationship between bases 2, 8,and 16 „^ Explain computing and bases that are powers of^2

CSE1520 W08 M. Mandelbaum

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Integers A natural number, a negative number, zeroExamples:

Numbers^ Rational Numbers^ An integer or the quotient of two integersExamples: -249, -1, 0, 3/7, -2/

CSE1520 W08 M. Mandelbaum

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How many ones are there in 642?

600 + 40 + 2? Or is it 384 + 32 + 2? Or maybe… 1536 + 64 + 2?

Natural Numbers

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Continuing with our example…642 in base 10

positional notation

is:

6 x 10

2 = 6 x 100

• 4 x 10

1 =^

4 x 10

• 2 x 10º =

2 x 1

= 642 in base 10

This number is in base 10

The power indicatesthe position ofthe number

Positional Notation

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d^ * Rn^

n-1^ + d

* Rn-^

n-2^ + ... + d

* R + d 2

1

As a formula: 642 is

+^2

R is the baseof the number^1

n is the number ofdigits in the number

d is the digit in thei

th^ positionin the number

Positional Notation

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Decimal is base 10 and has 10 digits:

Binary is base 2 and has 2 digits:

Binary^ For a number to exist in a given number system,the number system must include those digits. Forexample, the number 284 only exists in base 9 andhigher.

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How are digits in bases higher than 10represented?^ With distinct symbols for 10 and above.Base 16 has 16 digits:

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F

Bases Higher than 10

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What is the decimal equivalent of thehexadecimal number DEF?

D x 16

2 = 13 x 256 = 3328

• E x 16

1 = 14 x 16 = 224

• F x 16º = 15 x 1

= 15= 3567 in base 10

Converting Hexadecimal to DecimalRemember, the digits in base 16 are0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

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14

What is the decimal equivalent of the binarynumber 1101110?

1 x 2

6 = 1 x 64 = 64

• 1 x 2

5 = 1 x 32 = 32

• 0 x 2

4 = 0 x 16 = 0

• 1 x 2

3 = 1 x 8

• 1 x 2

2 = 1 x 4

• 1 x 2

1 = 1 x 2

• 0 x 2º = 0 x 1

= 0= 110 in base 10

Converting Binary to Decimal

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16

Remember borrowing?

Apply that concept

here:

Subtracting Binary Numbers

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17

10 12 1010

9 11 1001

8 10 100

7 7 111

6 6 110

5 5 101

4 4 100

3 3 011

2 2 010

1 1 001

0 0 000

Decimal Octal Binary

Power of 2 Number System

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19

• Groups of Four (from right)• Convert each group

1010 1011A

B

10101011 is AB in base 16

Converting Binary to Hexadecimal

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20

While the quotient is

not

zero

Divide the decimal number by the new baseMake the remainder the next digit to the left inthe answerReplace the original dividend with the quotient

Algorithm for converting base 10 to otherbases

Converting Decimal to Other Bases