Binary Conversions

Number systems Binary to decimal Decimal to binary

Binary Humor

There are 10 kinds of people in the world - those who understand binary and those who don't.

Numbering Systems

Base 10 or decimal numbering systemBase-10 numbering systems dictate that the

numbering scheme begins to repeat after the tenth digit (in our case, the number 9).

Zero is always the first number. When we count, we usually count "00, 01, 02,

03, 04, 05 , 06, 07, 08, 09, 10, 11, 12, ...“

Numbering Systems

Base 10 or decimal numbering systemEach digit to the left and right of the decimal

point is given a name which identifies that digit's placeholder.

Each placeholder is a multiple of ten.For now lets just consider positive numbers.

Numbering Systems - Base TenEach placeholder is a base of

ten. 10º = ones

Any number to the zero power is always equal to 1.

nº=1 10º=1

10¹ = tens Any number to the first

power is always equal itself. n¹=n 10¹=10

10² = hundreds 10³ = thousands

T

H

O

U

S

A

N

D

S

H

U

N

D

R

E

D

S

T

E

N

S

O

N

E

S

7 4 0 8

Numbering Systems – Base Ten

Arithmetic expression of 8 in 7408. Work right to left of decimal point. The ones position in expanded notation

calculating the exponent.10º*8=8 is the same as 1*8=8

Numbering Systems – Base Ten

Number 7 4 0 8

Position

Name

Thousands Hundreds Tens Ones

Exponential

Expression

10³*7 10²*4 10¹*0 10º*8

Calculated

Exponent

1000*7 100*4 10*0 1*8

Sum of the powers of ten.

1000*7 + 100*4 + 10*0 + 1*8 = 7408

Numbering Systems – Base two

Binary system is based on multiples of two. In binary numbering the numbering scheme

repeats after the second digit. Let's count to five in binary: “0000, 0001, 0010,

0011, 0100, 0101“ Binary numbering includes names for digit

placeholders.

Numbering Systems – Base two

Picture a odometer that is only capable of counting to two.

Numbering Systems – Base two

Binary placeholders Ones Twos Fours Eights Sixteen's Thirty-twos Sixty-fours

Decimal placeholders Ones Tens Hundreds Thousands Ten-thousands Hundred-thousands Millions

Numbering Systems – Base two

If the binary system is based on powers of 2, why is there still a "ones" position?

Remember: Anything to the zero power is always equal to 1.

In binary, the "ones" position is represented by the exponential expression 2º.

Convert Binary to Decimal

Sum of the powers of two. 8*1 + 4*1 + 2*0 + 1*1 = 13

Number 1 1 0 1

Position

Name

Eights Fours Twos Ones

Exponential

Expression

2³*1 2²*1 2¹*0 2º*1

Calculated

Exponent

8*1 4*1 2*0 1*1

Convert Binary to Decimal

Step 1 - Write the binary number in a row, separating the digits into columns.

Number 1 1 0 1

Convert Binary to Decimal

Step 2 - I want to decide whether each digit placeholder is "ON" or "OFF.“

"1" is "ON" and a "0" is "OFF.“ We don't have to calculate any digit placeholders that

are turned off.

Number 1 1 0 1

ON/OFF On On Off ON

Convert Binary to Decimal

Step 3 - Write the exponential expressions ("powers of two") that represent each placeholder and multiply each expression by 1.

We do this only for the placeholders that are turned ON. For the placeholders which are turned OFF, we simply bring down

the zero from the number itself

Number 1 1 0 1

ON/OFF On On Off ON

Exponential

Expression

2³*1 2²*1 0 2º*1

Convert Binary to Decimal

Step 4 - Calculate the exponents to get a simple multiplication expression for each placeholder.

Number 1 1 0 1

ON/OFF On On Off ON

Exponential

Expression

2³*1 2²*1 0 2º*1

Calculated

Exponent

8*1 4*1 0 1*1

Convert Binary to Decimal

Step 5 - Solve the multiplication expressions from step #4.

Number 1 1 0 1

ON/OFF On On Off ON

Exponential

Expression

2³*1 2²*1 0 2º*1

Calculated

Exponent

8*1 4*1 0 1*1

Solved Multiplication

8 4 0 1

Convert Binary to Decimal Step 6 - Add all the multiplication answers from

step #5 together to get our decimal number

Number 1 1 0 1

ON/OFF On On Off ON

Exponential

Expression

2³*1 2²*1 0 2º*1

Calculated

Exponent

8*1 4*1 0 1*1

Solved Multiplication

8 4 0 1

Add to calculate Value

8+4+0+1=13

Convert Binary to DecimalExample

Number 1 0 1 1 0 1

ON/OFF On Off On On Off On

Exponential

Expression

25 0 2³ 2² 0 2º*1

Calculated

Exponent

32*1 0 8*1 4*1 0 1*1

Solved Multiplication

32 0 8 4 0 1

Add to calculate

Value

32+0+8+4+0+1=45

Covert Decimal to Binary

Step 1 - Take the decimal number and divide it by 2.

Important: NEVER carry your divisions past the decimal point!

Decimal Number=97

Division Expression

Quotient Remainder

97/2 48 1

Covert Decimal to Binary

Step 2 - For each subsequent row, take the quotient from the previous row and divide it by two

Decimal Number=97

Division Expression Quotient Remainder

97/2 48 1

48/2 24 0

24/2 12 0

12/2 6 0

6/2 3 0

3/2 1 1

1/2 0 1

Covert Decimal to Binary

Step 3 – The remainder column only has ones or zeros.

The last cell in the remainder column of the last row must be a "1".

Read the 1s and 0s in the remainder column from the bottom to the top, we'll have our binary number!

Covert Decimal to Binary

Decimal Number=97

Division Expression

Quotient Remainder Direction

97/2 48 1

48/2 24 0

24/2 12 0

12/2 6 0

6/2 3 0

3/2 1 1

1/2 0 1

Binary Number=1100001

Read

Whiteboard Examples In Class Correction

37

DE Q R

37/2 18 1

18/2 9 0

9/2 4 1

4/2 2 0

2/2 1 0

1/2 0 1

1 0 0 1 0 1

25 24 23 22 21 20

32*1 16*0 8*0 4*1 2*0 1*1

32 0 0 4 0 1

32+0+0+4+0+1= 37

Read

The last cell in the remainder column of the last row must be a "1“ because we need to use whole numbers (nonnegative integers).1 ÷ 2 = 0 because 1 can not be divided into, 1 is the remainder.

37 (Odd Number)

DE Q R

37/2 18 1

18/2 9 0

9/2 4 1

4/2 2 0

2/2 1 0

1/2 0 1

36 (Even Number

DE Q R

36/2 18 0

18/2 9 0

9/2 4 1

4/2 2 0

2/2 1 0

1/2 0 1

Read

Read

Hexadecimal Conversation and ASCII

Hexa + Decimal

Base-16 number system It’s all Greek to me

“Sexa” = Latin = Six “Decimal” = Latin = Ten In 1963 IBM thought “Sexadecimal” was not politically

correct “Hexa” = Greek = Six Since the western alphabet contains only ten digits,

hexadecimal uses the letters A-F to represent the digits ten through fifteen.

Hexadecimal and Computing

It is much easier to work with large numbers using hexadecimal values than decimal or binary.One Hexadecimal digit = 4bitsTwo hexadecimal digits = 8 bitsEight bits=1 byteThis makes conversions between hexadecimal

and binary very easy

Counting Hexadecimal

Starting from zero, we count 00, 01, 02,03, 04, 05, 06, 07, 08, 09, 0A, 0B, 0C, 0D, 0E, 0F,10, 11, 12, 13, 14, 15, 16, 17 18, 19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21, 22, 23, 24, 25,....

Decimal Binary Hexadecimal

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 9

10 1010 a

11 1011 b

12 1100 c

13 1101 d

14 1110 e

15 1111 f

Convert Hexadecimal to Decimal

1 1 A =10 8

163 162 161 160

4096*1 256*1 16*10 8*1

4096 256 160 8

4096+256+160+8= 4520

Convert Decimal to Hexadecimal4520

DE Q R

4520/16 282 (.5*16)=8

282/16 17 (.625*16)=1010=A

17/16 1 (.0625*16)=1

1/16 0 (.0625*16)=1

11A8

Quotient must be a whole number. If decimal, multiply decimal portion by 16 for remainder. Remainder must be a whole number.

Read

Convert Hexadecimal to Binary

Convert each hexadecimal digit into its 4-bit binary equivalent.

1AB

Hex 1 A B

Bin 0001 1010 1011

000110101011

Convert Binary to Hexadecimal

Converteach 4bit binary digit into its hexadecimal equivalent starting from the right.

If there is an odd number of bits, add zeros to the left to make a complete 4bit digit.

110101011

Bin 0001 1010 1011

Hex 1 A B

1AB

Uses

Web pages http://www.psyclops.com/tools/rgb/

Networking MAC address

Programming C, C++, C#, Java, Assembly

Geeky T-shirts DEADB4C0FFEE

ASCII

American Standard Code for Information Interchange

Each character is 7bits + 1bit for parity = 1byte Represents English characters as numbers, with

each letter assigned a number from 0 to 127 This makes it possible to transfer data from one

computer to another. Used to store text files http://www.pcguide.com/res/tablesASCII-c.html http://nickciske.com/tools/binary.php

Conversion Lab Section I: Converting from Decimal to Binary

1) 11 2) 27 3) 54 4) 113 5) 273

Section II: Converting from Binary to Decimal 6) 101 7) 1011 8) 10100 9) 111010 10) 1010001

Conversion Lab

Section III: Convert Hexadecimal to Binary 11) 43B 12) DAB 13) 954 14) C0FFEE 15) B0A

Section IV: Convert Binary to Hexadecimal 16) 11000001111 17) 10100011110 18) 100110 19) 11011110 20) 101110110001

Conversion Lab Section V: Convert Hexadecimal to Decimal

21) FF2 22) 45 23) 19D 24) 345 25) AA

Section VI: Convert Decimal to Hexadecimal 26) 27 27) 85 28) 562 29) 4522 30) 5627