Chapter 2

Binary Values and Number Systems

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, 8, and 16

• Explain the importance to computing of bases that are powers of 2

2624

Numbers

32

Natural numbers, a.k.a. positive integersZero and any number obtained by repeatedly adding

one to it.

Examples: 100, 0, 45645, 32

Negative numbersA value less than 0, with a – sign

Examples: -24, -1, -45645, -32

43

IntegersA natural number, a negative number, zero

Examples: 249, 0, - 45645, - 32

Rational numbersAn integer or the quotient of two integers

Examples: -249, -1, 0, 3/7, -2/5

Real numbersIn general cannot be represented as the quotient of any

two integers. They have an infinite # of fractional digits.

Example: Pi = 3.14159265…

2.2 Positional notation

54

How many ones (units) are there in 642?

600 + 40 + 2 ?

Or is it

384 + 32 + 2 ?

Or maybe…

1536 + 64 + 2 ?

Positional Notation

65

Aha!

642 is 600 + 40 + 2 in BASE 10

The base of a number determines the number

of digits and the value of digit positions

Positional Notation

76

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

This number is in

base 10

The power indicates

the position of

the digit inside the

number

Positional Notation

87

dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1

As a formula:

642 is 63 * 102 + 42 * 10 + 21

R is the base

of the number

n is the number of

digits in the number

d is the digit in the

ith position

in the number

Positional Notation reloaded

97

dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1

In CS, binary digits are numbered from zero, to match

the power of the base:

dn-1 * Rn-1 + dn-2 * Rn-2 + ... + d1 * R1 + d0 * R0

dn-1 * 2n-1 + dn-2 * 2n-2 + ... + d1 * 21 + d0 * 20

Bit zero

(LSB)

Bit oneBit n-1

(MSB)

Positional Notation

1068

What if 642 has the base of 13?

642 in base 13 is equal to 1068 in base 10

64213 = 106810

+ 6 x 132 = 6 x 169 = 1014

+ 4 x 131 = 4 x 13 = 52

+ 2 x 13º = 2 x 1 = 2

= 1068 in base 10

Positional Notation

11

In a given base R, the digits range

from 0 up to R – 1

R itself cannot be a digit in base R

Trick problem:

Convert the number 473 from base 6 to base 10

Binary

129

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

13

Converting Binary to Decimal

14

What is the decimal equivalent of the binary

number 1101110?

11011102 = ???10

13

Converting Binary to Decimal

15

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

More practice with binary numbers:

100110102 = ???10

16

Bases Higher than 10

1710

How are digits in bases higher than 10

represented?

With distinct symbols for 10 and above.

Base 16 (hexadecimal, a.k.a. hex) has 16

digits:

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

Converting Hexadecimal to Decimal

18

What is the decimal equivalent of the

hexadecimal number DEF?

D x 162 = 13 x 256 = 3328

+ E x 161 = 14 x 16 = 224

+ F x 16º = 15 x 1 = 15

= 3567 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

More practice with hex numbers:

2AF16 = ???10

19

Converting Octal to Decimal

20

What is the decimal equivalent of the octal

number 642?

6428 = ???10

11

Converting Octal to Decimal

21

What is the decimal equivalent of the octal

number 642?

6 x 82 = 6 x 64 = 384

+ 4 x 81 = 4 x 8 = 32

+ 2 x 8º = 2 x 1 = 2

= 418 in base 10

11

Are there any non-positional number systems?

Hint: Why did the Roman civilization have nocontributions to mathematics?

22

Today we’ve covered pp.33-39 of the text (stopped before Arithmetic in Other Bases)

Solve in notebook for next class:1, 2, 3, 4, 5, 20, 21

No classes Monday!

23

1-minute quiz (in notebook)Convert to decimal:

1101 00112 = ???10

24

From the history of computing: bi-quinary

25

The front panel of the legendary

IBM 650 IBM 650 (source: Wikipedia)

Roman abacus (source: MathDaily.com)

Extra-credit question:Is bi-quinary a positional representation? Explain, either way.

26

Addition in Binary

27

Remember that there are only 2 digits in binary,

0 and 1

1 + 1 is 0 with a carry

Carry Values0 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

Addition in Binary

28

Practice addition:

Carry values

go here1 0 1 0 1 1 0

+1 0 0 0 0 1 1

14

Check in base ten!

Subtraction in Binary

29

Remember borrowing? Apply that concept

here:

1 2

0 2 0 2

1 0 1 0 1 1 1 1 0 1 0 1 1 1

- 1 1 1 0 1 1 - 1 1 1 0 1 1

0 0 1 1 1 0 0 0 0 1 1 1 0 0

15

Borrow values

Check in base ten!

Subtraction in Binary

30

Practice subtraction:

1 0 1 1 0 0 0

- 1 1 0 1 1 1

15

Borrow values

Check in base ten!

Converting Decimal to Other Bases

31

While (the quotient is not zero)Divide the decimal number by RMake 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 any other base R:

19

A.k.a. repeated division (by the base):

Converting Decimal to Binary

32

Example: Convert 17910 to binary

179 2 = 89 rem. 1

2 = 44 rem. 1

2 = 22 rem. 0

2 = 11 rem. 0

2 = 5 rem. 1

2 = 2 rem. 1

2 = 1 rem. 0

17910 = 101100112 2 = 0 rem. 1

Notes: The first bit obtained is the rightmost (a.k.a. LSB)

The algorithm stops when the quotient (not the remainder!)

becomes zero

19

LSBMSB

Converting Decimal to Binary

33

Practice: Convert 4210 to binary

42 2 = rem.

4210 = 2

19

Converting Decimal to Octal

34

What is 198810 in base 8?

Apply the repeated division algorithm!

Converting Decimal to Octal

35

248 31 3 0

8 1988 8 248 8 31 8 3

16 24 24 0

38 08 7 3

32 8

68 0

64

4

Answer is : 3 7 0 4

Converting Decimal to Hexadecimal

36

What is 356710 in base 16?

Work it out!

20

Converting Decimal to Hexadecimal

37

222 13 0

16 3567 16 222 16 13

32 16 0

36 62 13

32 48

47 14

32

15

D E F

21

Today we’ve covered:--pp.39-40 (Arithmetic in Other Bases)--pp.42-43 (Converting from base 10 to other bases).The section in between (Power of 2 number systems) will be covered next time, along with the rest of Ch.2.

Solve in notebook for next class:6, 7, 8, 9, 10, 11, 33a, 34a

38

Converting Binary to Octal

39

• Mark groups of three (from right)

• Convert each group

10101011 10 101 011

2 5 3

10101011 is 253 in base 8

17

Converting Binary to Hexadecimal

40

• Mark groups of four (from right)

• Convert each group

10101011 1010 1011

A B

10101011 is AB in base 16

18

Counting

41

Note the patterns!

42

On a new page in your notebook:

• Count from 0 to 31 in decimal

• Add the binary column

• Add the octal column

• Add the hex column

• Add the “base 5” (quinary) column

Converting Octal to Hexadecimal

43

End-of-chapter ex. 25:

Explain how base 8 and base 16 are related

10 101 011 1010 1011

2 5 3 A B

253 in base 8 = AB in base 16

18

Binary Numbers and Computers

44

Computers have storage units called binary digits or

bits

Low Voltage = 0

High Voltage = 1

All bits are either 0 or 1

22

Binary and Computers

45

Word= group of bits that the computer processes

at a time

The number of bits in a word determines the

word length of the computer. It is usually a

multiple of 8.

1 Byte = 8 bits

• 8, 16, 32, 64-bit computers

• 128? 256?

23

Read and take notes:Ethical Issues

46

Homeland Security

How does the Patriot Act affect

you?

your sister, the librarian?

your brother, the CEO of an ISP?

What is Carnivore?

Against whom is Carnivore used?

Has the status of the Patriot Act changed

in the last year?

Who am I?

47

Write three things

about me in your

notebook

Individual work(To do by next class, in notebook):

End-of-chapter questions 41-45

48

HomeworkTurn in next Monday, Sept. 20:

End-of-chapter exercises

23, 26, 28, 29, 30, 31, 38

Thought question 4