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### Chapter 2

Binary Values and Number Systems

- Eswari Manickam

Materials are from text book with additions and adaptations by Eswari Manickam

Chapter Goals

- Distinguish among categories of numbers
- Describe positional notation
- Convert numbers in other bases to base 10
- Convert base-10 numbers to numbers in other bases
- Describe the relationship between bases 2 and 16
- Explain the importance to computing of bases that are powers of 2

24

6

Binary

Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9

Binary is base 2 and has 2 digits:

0,1

For a number to exist in a given base, it can only contain the digits in that base, which range from 0 up to (but not including) the base.

What bases can these numbers be in? 122, 198, 178, G1A4

9

Positional Notation

- 642 is 600 + 40 + 2 in BASE 10

Continuing with our example…

642 in base 10 positional notation is:

6 x 102 = 6 x 100 = 600

+ 4 x 101 = 4 x 10 = 40

+ 2 x 10º = 2 x 1 = 2 = 642 in base 10

Converting Binary to Decimal

What is the decimal equivalent of the binary number 1101110?

1 x 26 = 1 x 64 = 64

+ 1 x 25 = 1 x 32 = 32

+ 0 x 24 = 0 x 16 = 0

+ 1 x 23 = 1 x 8 = 8

+ 1 x 22 = 1 x 4 = 4

+ 1 x 21 = 1 x 2 = 2

+ 0 x 2º = 0 x 1 = 0

= 110 in base 10

13

Converting Binary to Decimal

What is the decimal equivalent of the following binary numbers?

- a) 11101
- b) 1011010
- c) 10011100

Bases Higher than 10

How are digits in bases higher than 10 represented?

With distinct symbols for 10 and above.

Hexadecimal is base 16 and has 16 digits:

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F

10

Converting Hexadecimal to Decimal

What is the decimal equivalent of the hexadecimal number 32A?

3 x 162 = 3 x 256 = 768

+ 2 x 161 = 3 x 16 = 48

+ A x 16º = 10 x 1 = 10

= 826 in base 10

Remember, the digits in base 16 are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Converting Hexadecimal to Decimal

What is the decimal equivalent of the following hexadecimal numbers?

- 87A
- b) 34E
- c) F000

Converting Binary to Hexadecimal

- Mark groups of four (from right)
- Convert each group
- 10101011 10101011
- A B
- 10101011 is AB in base 16

18

Converting Binary to Hexadecimal

What is the hexadecimal equivalent of the following binary numbers?

- 00001001
- 10101001
- 010111011110

Converting Decimal to Other Bases

Algorithm for converting number in base 10 to other bases

- While (the quotient is not zero)
- Divide the decimal number by the new base
- Make the remainder the next digit to the left in the answer
- Replace the original decimal number with the quotient

19

Converting Decimal to Hexadecimal

What is the hexadecimal equivalent of (3567)10 ?

222 13 0

16 3567 16 222 16 13

3216 0

36 62 13

3248

47 14

32

15

D E F

21

Converting Decimal to Hexadecimal

What is 356 (base 10) in base 16?

What is 1135 (base 10) in base 16?

What is 4759 (base 10) in base 16?

Try it!

20

Converting Decimal to Binary

Example of converting decimal to binary

What is the binary equivalent of the decimal number 35?

17 8 4 2 1 0

2 35 2 17 2 8 4 2 1

3416842 0

11 00 0 1

Adding digits to the left as we calculate: 100011

Converting Decimal to Binary

Easy method for converting decimal to binary

What is the binary equivalent of the decimal number 35?

2 35 - 1

2 17 - 1

2 8 - 0

2 4 - 0

2 2 - 0

2 1 - 1

0

So reading from the bottom – The answer would be 100011

Converting Decimal to Binary

What is the binary equivalent of the following decimal integers?

A) 64

B) 1066

C) 213

D) 1790

Arithmetic in Binary

Remember that there are only 2 digits in binary, 0 and 1

1 + 1 is 0 with a carry

Carry Values

1 1 1 1 1 1

1 0 1 0 1 1 1

+1 0 0 1 0 1 1

1 0 1 0 0 0 1 0

14

Subtracting Binary Numbers

Remember borrowing? Apply that concept here:

1 2

2 0 2

1 0 1 0 1 1 1

- 1 1 1 0 1 1

0 0 1 1 1 0 0

15

Binary Numbers and Computers

Computers have storage units called binary digits or bits

Low Voltage = 0

High Voltage = 1 all bits have 0 or 1

22

Binary and Computers

- Byte
- 8 bits
- The number of bits in a word determines the word length of the computer, but it is usually a multiple of 8
- 32-bit machines
- 64-bit machines etc.

23

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