number systems different number systems representation of numbers in binary conversion between...

Number Systems Different number systems Representation of numbers in binaryConversion between decimal and binary,Conversion between binary and hexadecimalUse of subscripts 2, 10 and 16 for bases

Number Systems Decimal number system – Base

10 = 1, 2 ,3 4, 5, ect..

Binary number system –Base 2 = 0001, 0010, 0011, ect…

Hexadecimal number system = Base 16 = 9, A, B, 4C ect…

Decimal Number Systems

Hundreds Tens Units

341

102 101 100

300 40 1

300 + 40 +1 = 341

Decimal numbers are base 10

They are made up of 10 numbers – 0,1,2,3,4,5,6,7,8,9.

Combining the ten numbers will create units, tens, hundreds and thousands

Split the following decimal numbers

Hundreds Tens Units

550

Hundreds Tens Units

982

AnswersHundreds Tens Units

550

102 101 100

500 50 0

500 + 50 + 0 = 55010

Hundreds Tens Units

982

102 101 100

900 80 2

900 + 80 + 2 = 98210

Binary Number System Binary numbers are base 2Computer languageThey are made up of 2 numbers –

1 and 0Decimal Binary Decimal Binary

010 02 510 1012

110 12 610 1102

210 102 710 1112

310 112 810 10002

410 1002 910 10012

Hexadecimal Number Systems Hexadecimal numbers are base

16Computer memory locationsThey are made up of 16 numbers

Decimal Hex Decimal Hex

010 016 510 516

110 116 610 616

210 216 710 716

310 316 810 816

410 416 910 916

Decimal Hex

1010 A16

1110 B16

1210 C16

1310 D16

1410 E16

Decimal Hex

1510 F16

Importance of Base numbers Writing the base numbers is very

important as;

◦1510 and 1516 are not the same number but without the base they would be both considered as the same number

◦1010 and 102 are not the same number as 102 represents 210

Complete the table

Number Number System

2010

2A16

10101012

10110

1516

1110001112

Answers

Number Number System

2010 Decimal

2A16 Hexadecimal

10101012 Binary

10110 Decimal

1516 Hexadecimal

1110001112

Binary

Converting Binary to Decimal

Explanation

1. Write down the placement value on top of each number.

2. Write the values that are on (the ones with a one under them

3. Add the numbers together

24 23 22 21 20

16 8 4 2 1

Example We want to convert 110012 to

decimal 24 23 22 21 20

1 1 0 0 1

16 8 4 2 1

16 8 1

16 + 8 + 1

25

Working Convert the following to decimal

1. 1010102

2. 1110112

3. 101010012

4. 0011001112

5. 1110101002

AnswersConvert the following to decimal

1. 1010102 = 4210

2. 1110112 = 5910

3. 101010012 = 16910

4. 0011001112 = 10310

5. 1110101002 = 46810

Converting Decimal to Binary

Method One

1. Write down the placement values of binary

2. Chose the numbers that add up to you decimal number

3. Put a 1 under the numbers used to add up to your decimal number

124

64 32 16 8 4 2 1

Example Convert 4610 to binary

124

64 32 16 8 4 2 1

0 0 1 0 1 1 1 0

32 + 8 + 4 + 2 = 46

4610 = 001011102

Method TwoDivide the original number by 2

and write down the remainder even if it is 0

Keep on dividing the decimal numbers by 2 until 1 is divided by 2

Write down the remainders next to each other starting from the bottom moving upwards

Example Convert 4610 to binary

Ans 4610 = 1011102

46 / 2 = 23 r 0

23 / 2 = 11 r 1

11 / 2 = 5 r 1

5 / 2 = 2 r 1

2 / 2 = 1 r 0

1 / 2 = 0 r 1

Working Convert the following decimal

numbers to binary 1. 1010

2. 6610

3. 12010

4. 3510

5. 8810

AnswersConvert the following decimal

numbers to binary 1. 1010 = 10102

2. 6610 = 10000102

3. 12010 = 11110002

4. 3510 = 1000112

5. 8810 = 10110002

Converting Binary to Hexadecimal

ExplanationSplit the binary number into

groups of 41001110 = 0100 – 1110

Write the 2x on top of each number starting from the right

Add the numbers that are on Write down the totals, if a total is

larger than 9, convert it to the hex letter

0 1 0 0 1 1 1 023

22

21 20

23

22

21

20

8 4 2 1 8 4 2 14 14

4E16

NOTE: when we do not have enough bits lefts to create a group of 4 we add 0s

ExampleConvert 11001112 in

Hexadecinal 0 1 1 0 0 1 1 1

23 22 21 20 23 22 21 20

8 4 2 1 8 4 2 1

6 7

6716

Working Convert the following into

Hexadecimal

1. 1110101002

2. 11101112

3. 1010102

4. 1112

5. 11100012

Working Convert the following into

Hexadecimal

1. 1110101002 = 1D416

2. 11101112 = 7716

3. 1010102 = 2A16

4. 1112 = 716

5. 11100012 = 7116

Converting Hexadecimal to Binary

Explanation

1. Write each individual number in the hexadecimal number eg B416

2. Write the binary placement values for each hex number

3. List 1s under the placement values that are onB = 11 4

23

22 21 20 23 22 21 20

8 4 2 1 8 4 2 11 0 1 1 0 1 0 0

101101002

4. Write the split binary number as one whole number

ExampleConvert 2C16 to binary

2 C = 12

23 22 21 20 23 22 21 20

8 4 2 1 8 4 2 1

0 0 1 0 1 1 0 0

001011002

Working Convert the following hex

numbers to binary

1. AB16

2. F716

3. 1516

4. CC16

5. 2216

AnswersConvert the following hex

numbers to binary

1. AB16 = 101010112

2. F716 = 111101112

3. 1516 = 000101012

4. CC16 = 110011002

5. 2216 = 001000102

Converting Decimal to Hexadecimal

Method OneDivide the decimal number by 16

taking note of the remaindersKeep on dividing the whole

number by 16 until the whole number obtained is 0.

Write down the remainders next to each other starting from the bottom, changing numbers greater than 9 to letters

46

5

/ 16 = 29 r 1

29 / 16 = 1 r 13

1 / 16 = 0 r 1

ANS = 1D116

Example Convert 80010 to hexadecimal

80

0

/ 16 = 50 r 0

50 / 16 = 3 r 2

3 / 16 = 0 r 3

ANS = 32016

Method Two

1. Convert the decimal number to binary

2. Convert the binary number to hexadecimal

Eg, changing 45610 to hexadecimal

Example Convert 80010 to hexadecimal

512

256

128

64 32 16 8 4 2 1

1 1 0 0 1 0 0 0 0 0

512 + 256 + 32 = 800

80010 = 110010000020 0 1 1 0 0 1 0 0 0 0 023 22 21 20 23 22 2

1

20 23 22 21 20

8 4 2 1 8 4 2 1 8 4 2 13 2 0

32016

WorkingConvert the following to

Hexadecimal numbers1. 34010

2. 11910

3. 6610

4. 2510

5. 11110

AnswersConvert the following to

Hexadecimal numbers1. 34010 = 15416

2. 11910 = 7716

3. 6610 = 4216

4. 2510 = 1916

5. 11110 = 6F16

Converting Hexadecimal to

Decimal

ExplanationWriting down the placement

values on top of each number starting with 160

Multiply the top value with the hexadecimal number.

Add all the results162

256

161

16

160

1

4 3 A

(256x4) (16x3) (1x10)

1024 48 10

=1024+48+10

=108210

Converting 43A16 to decimal

WorkingConvert the following into

decimal

1. 5516

2. CB16

3. F816

4. B416

5. 9016

AnswersConvert the following into

decimal

1. 5516 = 8510

2. B016 = 17610

3. 2F816 = 76010

4. B416 = 18010

5. 9016 = 14410

Homework Copy and complete this table

Decimal Binary Hexadecimal

2110

1010101002

2E16

15910

001110002

1C216

4410