Chapter 2

Binary Values and Number Systems

- Eswari Manickam

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

2 624

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

3 9

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

Binary

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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

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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

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Converting Binary to Decimal

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Converting Binary to Decimal

What is the decimal equivalent of the following binary numbers?

a) 11101

b) 1011010

c) 10011100

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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

Bases Higher than 10

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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

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Converting Hexadecimal to Decimal

What is the decimal equivalent of the following hexadecimal numbers?

a) 87A

b) 34E

c) F000

10

Counting in Binary/Hexadecimal/DecimalDecimal Hexadecimal Binary

0 0 0000

1 1 0001

2 2 0010

3 3 0011

4 4 0100

5 5 0101

6 6 0110

7 7 0111

8 8 1000

9 9 1001

10 A 1010

11 B 1011

12 C 1100

13 D 1101

14 E 1110

15 F 1111

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• Mark groups of four (from right)• Convert each group

10101011 1010 1011 A B

10101011 is AB in base 16

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Converting Binary to Hexadecimal

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Converting Binary to Hexadecimal

What is the hexadecimal equivalent of the following binary numbers?

• 00001001

• 10101001

• 010111011110

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While (the quotient is not zero)Divide the decimal number by the new baseMake the remainder the next digit to the left in the

answerReplace the original decimal number with the quotient

Algorithm for converting number in base 10 to other bases

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Converting Decimal to Other Bases

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222 13 0 16 3567 16 222 16 13

32 16 0 36 62 13 32 48 47 14 32 15

D E F

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Converting Decimal to Hexadecimal

What is the hexadecimal equivalent of (3567)10 ?

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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!

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Converting Decimal to Hexadecimal

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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

34 16 8 4 2 0 1 1 0 0 0 1

Adding digits to the left as we calculate: 100011

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Converting Decimal to Binary

Easy method for converting decimal to binaryWhat 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

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Converting Decimal to Binary

What is the binary equivalent of the following decimal integers?

A) 64

B) 1066

C) 213

D) 1790

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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

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Arithmetic in Binary

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Arithmetic in Binary

Calculate:

a) 10001 + 11101

b) 1110 + 1111

c) 1011001 + 111010

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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

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Subtracting Binary Numbers

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Subtracting Binary Numbers

Calculate:

A) 1011011 - 10010

B) 1010110 - 101010

C) 1000101 - 101100

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Computers have storage units called binary digits or bits

Low Voltage = 0High Voltage = 1 all bits have 0 or 1

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Binary Numbers and Computers

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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.

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Binary and Computers