NUMBERING SYSTEMS shared.ppsx

8/10/2019 NUMBERING SYSTEMS shared.ppsx

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A binary numberis a number expressed in

the binary numeral system, or base-2

numeral system, which represents numeric

values using two different symbols: typically

0 (zero) and 1 (one)

More specifically, the usual base-2 system is

a positional notation with a radix of 2

http://en.wikipedia.org/wiki/Binary_number


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According to wikipedia.com, octal numeral

system is the base-8 number system, and

uses the digits 0 to 7.

http://en.wikipedia.org/wiki/Octal
http://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Radixhttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Numeral_systemhttp://en.wikipedia.org/wiki/Numeral_system


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The decimalnumeral system has tenas its

base.

It is the numerical base most widely used bymodern civilizations.

http://en.wikipedia.org/wiki/Decimal


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According to wikipedia.com, hexadecimal

(also base 16, or hex) is a positional

numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often

the symbols 09to represent values zero to

nine, and A,B,C,D,E,F(or alternatively a

f) to represent values ten to fifteen.

http://en.wikipedia.org/wiki/Hexadecima


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67 mod 2 = 1

33 mod 2 = 1

16 mod 2 = 0

8 mod 2 = 0

4 mod 2 = 0

2 mod 2 = 0

1

Result :

6710 = 10000112

6710= ..... 2

R

E

AD

D

I

V


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67 mod 8 = 3

8 mod 8 = 0

1

Result :

6710 = 1038

6710= ..... 8

R

E

AD

D

I

V


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624 mod 16 = 0

39 mod 16 = 7

2 mod 16 = 2

0

Result :

6710 = 27016

62410= ..... 16

R

E

AD

D

I

V


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1011102 = . 10(1x 25) + (0x 24) + (1x 23) + (1x 22) + (1x 21) + (0x 20

32 + 0 + 8 + 4 + 2 + 0

4610


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7648 = . 10(7x 82) + (6x 81) + (4x 80)

50010

448 + 48 + 4


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F516= .10(15(F)x 161) + (5x 160)

24510

240 + 5


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/Addition/index.ht


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/Subtraction/ind


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/Multiplication/inde


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/Division/index.htm


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When our number ispositive, we make oursign bit zero,and whenour number is negative,we make our sign bitone.

This approach is calledthe Signed MagnitudeRepresentation.

Weakness : We need to specify how

many bitsare in ournumbers so we can becertain which bit isrepresenting the sign.

Example: First, we convert 5 to

binary.

101 (5)

Now we add a sign bit.Notice that we havepadded '1' with zeros soit will have four bits.

0101 (5)

To make our binarynumbers negative, wesimply change our signbit from '0' to '1'.

1101 (-5)

http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/SignedNumbers/index


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Here is a quick

summary of how to

find the 1's

complementrepresentation of any

decimal numberx.

Ifxis positive, simply

convertxto binary. Ifxis negative, write

the positive value ofx

in binary

Reverse each bit.

Example:

First, we write the

positive value of the

number in binary. 0101 (+5)

Next, we reverse

each bit of the

number so 1'sbecome 0's and 0's

become 1's

1010 (-5)

http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/OnesComplement/inde


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/SubtractionWithOnesComplement/inde


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/SubtractionWithOnesComplement/inde


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Here is a quick summaryof how to find the 2'scomplementrepresentation of any

decimal numberx. Ifxis positive, simply

convertxto binary.

Ifxis negative, write thepositive value ofxin

binary Reverse each bit.

Add 1 to thecomplemented number.

Example:

First, we write thepositive value of thenumber in binary.

0101 (+5) Next, we reverse each

bit to get the 1'scomplement. 1010

Last, we add 1 to thenumber. 1011 (-5)

http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/TwosComplement/inde


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http://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/SubtractionWithTwosComplement/inde

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