Binary Conversion

transcript

Binary

In 1854, British mathematician In 1854, British mathematician George BooleGeorge Boole published a paper published a paper

detailing a system of logic that would detailing a system of logic that would become known as become known as Boolean algebraBoolean algebra..

His logical system proved instrumental His logical system proved instrumental in the development of the binary in the development of the binary

system, particularly in its system, particularly in its implementation in electronic circuitry.implementation in electronic circuitry.

BinaryA numbering systems that only uses two A numbering systems that only uses two

digits. digits. 00 and and 11..Rather than a base ten that we are all Rather than a base ten that we are all

familiar with.familiar with. Computers use binary to store Computers use binary to store information in a digital format.information in a digital format.

Each digit ( Each digit ( 00 or or 11) represents one bit) represents one bitEight bits are equal to one byte.Eight bits are equal to one byte.

One Binary Digit abbreviation is “b”

Can be thought of as one character Either a 1 or a 0

Eight bits make up one byte Abbreviation “B” Combination of 1’s and 0’s Can be thought of as one character

11101010

kilobit

1024 bits Abbreviation “Kb”

kilobytes

Represented by KB Slang “Kilo” Is equal to 1024 bytes 210

megabytes

Represented by MB Slang “Meg” Is equal to 1,000000 bytes One million bytes 220

gigabyte

Represented by GB Slang “Gig” Equal to 1,000,000,000 Bytes One Billion bytes 230

terabyte

Represented by TB Slang “tera” Equal to 1,000,000,000,000 Bytes One Trillion bytes 240

petabyte

Represented by PB Slang “peta” Equal to 1,000,000,000,000,000

Bytes One Thousand Trillion bytes 250

exabyte

Represented by EB Slang “exa” Equal to

1,000,000,000,000,000,000 Bytes

One Million Trillion bytes 260

All printed materialin the world would use about 5 Exabytes

Think of Binary as light bulbs

that are either ON or Off

All eight of these Light bulbs would represent one byte

One Light bulb represents one One Light bulb represents one BitBit

11 00 00 00 00 00 00 11

Binary ExerciseBinary Exercise

Bit Postion Bit 8 Bit 7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Total Binary ValuePosition Value

if ON128 64 32 16 8 4 2 1 255

Position Valueif OFF

0 0 0 0 0 0 0 0 0

Turn a Postion ON

1 0 0 0 0 0 0 1

Here we would ADD

The Postion Value

Here we bringThe Postion

Value DOWN

128 1 129

In this incidence our binary Number 10000001 would have a value of 129Because Postion 8 is ON Postion 7 is OFF Postion 6 is OFF Position 5 is OFF Position 4 is Off

Postion 3 is OFF Postion 2 is OFF and Position 1 is ON.

Binary Exercise

Binary

Figuring Binary. Starting on the right going to the left The first digit will be 1 The second digit will be 2 The third digit will be 4 The fourth digit will be 8 The fifth digit will be 16 The sixth digit will be 32 The seventh digit will be 64 The eighth digit will be 128

Binary

Base Ten numbers are tabulated Left to Right.

Binary

Binary numbers are tabulatedRight to Left.

Example10000000The 1st – 7th digit would be OffThe Eighth digit would be On

The first digit will be 1 0 The second digit will be 2 0 The third digit will be 4 0 The fourth digit will be 8 0 The fifth digit will be 16 0 The sixth digit will be 32 0 The seventh digit will be 64 0 The eighth digit will be 128 +128

Add the bits

The value of the number would be Total 128

Example10000001The 1st digit would be OnThe 2nd – 7th digit would be OffThe Eighth digit would be On

Add the bits

Example10000011The 1st digit would be OnThe 2nd digit would be OnThe 3rd – 7th digit would be OffThe Eighth digit would be On

Add the bits The value of the number would be Total 131

Example10000111The 1st- 3rd digit would be OnThe 4th – 7th digit would be OffThe Eighth digit would be On

The first digit will be 1 1 The second digit will be 2

2 The third digit will be 4 4 The fourth digit will be 8 0 The fifth digit will be 16 0 The sixth digit will be 32 0 The seventh digit will be 64

0 The eighth digit will be 128 +128

Add the bits

Example11000000The 1st- 6th digit would be OffThe 7th digit would be OnThe 8th digit would be On

0 The third digit will be 4 0 The fourth digit will be 8 0 The fifth digit will be 16 0 The sixth digit will be 32 0 The seventh digit will be 64

Add the bits

11 11 00 00 00 00 00 00

What is the What is the value?value?

192192

Example11111111The 1st- 8th digit would be On

2 The third digit will be 4

4 The fourth digit will be 8 8 The fifth digit will be 16 16 The sixth digit will be 32 32 The seventh digit will be 64

Add the bits

11 11 11 11 11 11 11 11

255255

128128 6464 3232 1616 88 44 22 11

Using Calculatorto figureBinary Numbers

First we would open Calculator Start/All Programs/Accessories/CalculatorFrom the Calculator go to View and down To SCIENTIFIC

Scientific

This is the Scientific Calculator The next thing we would need to do

in select BIN for Binary

Next we would enter the Binary number

For example

10000000

After entering the Binary number we would then select the

Dec Radio Button

We now see the answer to the problemIs

11 11 00 00 00 00 00 00

192192

Decimal to Binary

It follows a starightforward method. It involves dividing the number to be

converted, say N by 2 (since binary is in base 2) until we reach the division of (1/2), also making note of all remainders.

Example 1: Convert 98 from decimal to binary Divide 98 by 2, make note of all the

remainder. Continue dividingquotientsby 2,

making notes of the remainder. Also, note the star beside the last

remainder.

Division Remainder, R

98/2 = 49 R=0

49/2 = 24 R=1

24/2 = 12 R=0

12/2 = 6 R=0

6/2 = 3 R=0

3/2 = 1 R=1

1/2 = 0 R=1

The sequance of remainders going up gives the answer. Starting from 1*, we have 1100010.Therefore, 98 in decimals is 1100010 in binary

Example 2: Convert 21 into binary

Division Remainder, R

21/2 = 10 R=1

10/2 = 5 R=0

5/2 = 2 R=1

2/2 = 1 R=0

1/2 = 0 R=1

Therefore, 21 in decimals is 10101 in binary

Binary to decimal

Conversion follows the same steps as decimal to binary, except in reverse order.

We can begin by multiplying 0 x 2 and adding 1.

We continue to multiply the numbers in the “results” column by 2, and adding the digits from left to right in our binary numbers.

Example 1: Convert 11101 from binary to decimal

Operations Result

0 x 2 + 1 1

1 x 2 + 1 3

3 x 2 + 1 7

7 x 2 + 0 14

14 x 2 + 1 29

Therefore, 11101 in binary is 29 in decimal.

Binary Conversion

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