Binary Conversions

1
Binary Conversions

• Number systems
• Binary to decimal
• Decimal to binary

2
Binary Humor

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

3
Numbering Systems

• Base 10 or decimal numbering system
• Base-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, ...

4
Numbering Systems

• Base 10 or decimal numbering system
• Each 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.

5
Numbering Systems - Base Ten
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

• Each placeholder is a base of ten.
• 10º ones
• Any number to the zero power is always equal to
  1.
• nº1
• 10º1
• 10¹ tens
• Any number to the first power is always equal
  itself.
• n¹n
• 10¹10
• 10² hundreds
• 10³ thousands

6
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.
• 10º88 is the same as 188

7
Numbering Systems Base Ten
Number 7 4 0 8
Position Name Thousands Hundreds Tens Ones
Exponential Expression 10³7 10²4 10¹0 10º8
Calculated Exponent 10007 1004 100 18

• Sum of the powers of ten.
• 10007 1004 100 18 7408

8
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.

9
Numbering Systems Base two

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

10
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

11
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.

12
Convert Binary to Decimal
Number 1 1 0 1
Position Name Eights Fours Twos Ones
Exponential Expression 2³1 2²1 2¹0 2º1
Calculated Exponent 81 41 20 11

• Sum of the powers of two.
• 81 41 20 11 13

13
Convert Binary to Decimal

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

Number 1 1 0 1
14
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
15
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 2³1 2²1 0 2º1
16
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 2³1 2²1 0 2º1
Calculated Exponent 81 41 0 11
17
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 2³1 2²1 0 2º1
Calculated Exponent 81 41 0 11
Solved Multiplication 8 4 0 1
18
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 2³1 2²1 0 2º1
Calculated Exponent 81 41 0 11
Solved Multiplication 8 4 0 1
Add to calculate Value 840113 840113 840113 840113
19
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 2º1
Calculated Exponent 321 0 81 41 0 11
Solved Multiplication 32 0 8 4 0 1
Add to calculate Value 320840145 320840145 320840145 320840145 320840145 320840145
20
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 Number97 Decimal Number97 Decimal Number97
Division Expression Quotient Remainder
97/2 48 1
21
Covert Decimal to Binary

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

Decimal Number97 Decimal Number97 Decimal Number97
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
22
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!

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Covert Decimal to Binary
Decimal Number97 Decimal Number97 Decimal Number97 Decimal Number97
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 Number1100001 Binary Number1100001 Binary Number1100001 Binary Number1100001
Read
24
Whiteboard Examples In Class Correction
37 37 37 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
321 160 80 41 20 11
32 0 0 4 0 1
3200401 37 3200401 37 3200401 37 3200401 37 3200401 37 3200401 37
Read
25
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) 37 (Odd Number) 37 (Odd Number) 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 36 (Even Number 36 (Even Number 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
26
Hexadecimal Conversation and ASCII
27
Hexa Decimal

• Base-16 number system
• Its 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.

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Hexadecimal and Computing

• It is much easier to work with large numbers
  using hexadecimal values than decimal or binary.
• One Hexadecimal digit 4bits
• Two hexadecimal digits 8 bits
• Eight bits1 byte
• This makes conversions between hexadecimal and
  binary very easy

29
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,....

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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
31
Convert Hexadecimal to Decimal
1 1 A 10 8
163 162 161 160
40961 2561 1610 81
4096 256 160 8
40962561608 4520 40962561608 4520 40962561608 4520 40962561608 4520
32
Convert Decimal to Hexadecimal
4520 4520 4520 4520
DE Q R
4520/16 282 (.516)8
282/16 17 (.62516)1010A
17/16 1 (.062516)1
1/16 0 (.062516)1
11A8 11A8 11A8
Read

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

33
Convert Hexadecimal to Binary
Hex 1 A B
Bin 0001 1010 1011
000110101011 000110101011 000110101011 000110101011

• Convert each hexadecimal digit into its 4-bit
  binary equivalent.
• 1AB

34
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 1AB 1AB 1AB
35
Uses

• Web pages
• http//www.psyclops.com/tools/rgb/
• Networking
• MAC address
• Programming
• C, C, C, Java, Assembly
• Geeky T-shirts
• DEADB4C0FFEE

36
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

37
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

38
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

39
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