BINARY 1

Number Systems• Base 10

• uses the numbers 0-9• Represents numbers as ones, tens, hundreds etc

Hundreds Tens Ones Solution

0 4 3 0+40+3= 43

5 9 4 500+90+4= 594

Number Systems• Base 2

• uses the numbers 0-1• Each column is a factor of two• To calculate a conversion we add together the things that are

included (1) and not the things that aren’t (0)

128 64

32

16

8 4 2 1 Solution

0 0 0 1 0 0 0 1

0 1 0 1 0 0 1 0

1 0 0 0 0 1 0 0

1 1 0 0 1 0 0 1

128 64 32 16 8 4 2 1 Solution

0 0 0 1 0 0 0 1 16 + 1 = 17

0 1 0 1 0 0 1 0 64 + 16 + 2 = 82

1 0 0 0 0 1 0 0 128 + 4 = 132

1 1 0 0 1 0 0 1 128 + 64 + 8 + 1 = 201

Binary to Decimal Conversion• Practise converting the following binary to decimal

128 64 32 16 8 4 2 1 Solution

0 0 1 0 1 1 0 0

0 1 0 1 1 0 0 0

1 0 1 0 0 0 1 1

1 1 0 0 0 0 1 1

1 1 1 1 1 0 0 0

0 1 1 1 0 0 0 1

0 1 0 1 0 1 0 1

0 1 1 0 0 1 1 0

1 1 1 1 1 1 1 1

Binary to Decimal Conversion• SOLUTIONS

128 64 32 16 8 4 2 1 Solution

0 0 1 0 1 1 0 0 32 + 8 + 4 = 44

0 1 0 1 1 0 0 0 64 + 16 + 8 = 88

1 0 1 0 0 0 1 1 128 + 32 + 2 + 1 = 163

1 1 0 0 0 0 1 1 128 + 64 +2 + 1 = 195

1 1 1 1 1 0 0 0 128 + 64 + 32 + 16 + 8 = 248

0 1 1 1 0 0 0 1 64 + 32 + 16 + 1 = 113

0 1 0 1 0 1 0 1 64 + 16 + 4 + 1 = 85

0 1 1 0 0 1 1 0 64 + 32 + 4 + 2 = 102

1 1 1 1 1 1 1 1 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255

Decimal to Binary Conversion• To convert the other way, we work from the highest number to the

lowest number asking if it fits in our decimal number• Convert 39

• Does 128 fit into 39? No 0• Does 64 fit into 39? No 0• Does 32 fit into 39? Yes 1

how many are left over? 39-32 = 7• Does 16 fit into 7? No 0• Does 8 fit into 7? No 0• Does 4 fit into 7? Yes 1

how many are left over? 7-4 = 3• Does 2 fit into 3? Yes 1

how many are left over? 3-2 = 1• Does 1 fit into 1? Yes 1

how many are left over? 1-1=0

• 39 is 00100111

Decimal to Binary Conversion• Using the method in the previous slide, convert the

following decimal solutions to binary

128 64 32 16 8 4 2 1 Solution

5

19

27

35

49

87

167

232

240

Decimal to Binary Conversion• SOLUTIONS

128 64 32 16 8 4 2 1 Solution

0 0 0 0 0 1 0 1 5

0 0 0 1 0 0 1 1 19

0 0 0 1 1 0 1 1 27

0 0 1 0 0 0 1 1 35

0 0 1 1 0 0 0 1 49

0 1 0 1 0 1 1 1 87

1 0 1 0 0 1 1 1 167

1 1 1 0 1 0 0 0 232

1 1 1 1 0 0 0 0 240

Fractional Numbers• If a decimal number has values after the decimal point we

can still convert using the same process

2 1 . .5

.25

.125 .0625 .03125

.015625 Solution

0 1 . 1 0 0 0 0 0

1 0 . 0 1 1 0 0 0

0 0 . 0 1 1 1 0 0

1 1 . 1 0 0 1 0 1

2 1 . .5 .25 .125 .0625 .03125 .015625 Solution

0 1 . 1 0 0 0 0 0 1 + 0.5 = 1.5

1 0 . 0 1 1 0 0 0 2 + 0.25 + 0.125 = 2.375

0 0 . 0 1 1 1 0 0 0.25 + 0.125 + 0.0625 = 0.4375

1 1 . 1 0 0 1 0 1 2 + 1 + 0.5 + 0.0625 + 0.015625 = 3.578125

Fractional Numbers• Sometimes we find numbers that we can’t easily fit into

the binary grid – for these we use another system• Division by 2

• We repeatedly divide a number by two to get the final solution

Fractional Numbers – Division by 2• Convert 0.4

• Multiply it by 2• Put the whole number in one

column• Put the remainder in the next

column• Carry your remainder to be the

starting number in the next row• Repeat these steps• There is no rule how many

times you should do this... But I accept 5 times

Number Multiplic-ation

Whole Number

Part number

0.4 x 2 = 0 .8

0.8 x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

• The binary number in the ‘whole number’ column, from top to bottom, is your binary solution

Fractional Numbers – Division by 2

• Convert 0.3

Number Multiplic-ation

Whole Number

Part number

0.3 x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

• Convert 0.24

Number Multiplic-ation

Whole Number

Part number

0.24 x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

x 2 =

Characters using Binary• When we use 8 bit binary, each combination can be

converted into decimal.• Each decimal value has a character associated with it.• For example: A is 65

a is 97• Your ASCII character set has been provided as a separate

handout.

Characters using Binary• Convert the following:

Character Decimal

#

100

F

84

p

32

Characters using Binary• Solutions

Character Decimal

# 35

d 100

F 70

T 84

p 112

(space) 32

Worksheet• Test your new binary < -- > decimal conversion skills with

worksheet : Binary_1