Chapter 2

Binary Values and Number Systems

Numbers

2 2

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

one to it.

Examples: 100, 0, 45645, 32

Negative numbers A value less than 0, with a – sign

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

3 3

Integers A natural number, a negative number, zero

Examples: 249, 0, - 45645, - 32

Rational numbers An integer or the quotient of two integers

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

Real numbers In 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

4 4

How many ones (units) are there in 642?

600 + 40 + 2

6 x 100 + 4 x 10 + 2 x 1

6 x 102 + 4 x 101 + 2 x 100

10 is called the base

We also write 64210

QUIZ

5 4

How many ones (units) are there in 6429

(642 in base nine)?

Use the base-10 model:

600 + 40 + 2

6 x 100 + 4 x 10 + 2 x 1

6 x 102 + 4 x 101 + 2 x 100

QUIZ

6 4

How many ones (units) are there in 6429

(642 in base nine)?

6 x 92 + 4 x 91 + 2 x 90 = 12510

Positional Notation

7 5

The base of a number determines how many

digits are used and the value of each digit’s

position.

To be specific:

• In base R, there are R digits, from 0 to R-1

• The positions have for values the powers of

R, from right to left: R0, R1, R2, …

Positional Notation

8 7

dn * Rn-1 + dn-1 * R

n-2 + ... + d2 * R + d1

Formula:

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

9 7

dn * Rn-1 + dn-1 * R

n-2 + ... + d2 * R + d1

… but, in CS, the digits are numbered from zero, to

match the power of the base:

dn-1 * Rn-1 + dn-2 * R

n-2 + ... + d1 * R1 + d0 * R

0

The text shows the digits numbered like this:

QUIZ

10 6 8

What is 642 in base 13?

QUIZ

11 6 8

What is 642 in base 13?

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

Nota bene!

12

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

13 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

QUIZ: Converting Binary to Decimal

14

What is the decimal equivalent of the binary

number 110 1110?

110 11102 = ???10

13

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

= 11010

13

Base 8 = octal system

16

What is the decimal equivalent of the octal

number 642?

6428 = ???10

11

Converting Octal to Decimal

17

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

Add the above = 418 in base 10

= 41810

11

Bases Higher than 10

18 10

How are digits in bases higher than 10

represented?

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, F

Converting Hexadecimal to Decimal

19

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

QUIZ: 2AF16 = ???10

20

Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics?

21

Today we’ve covered pp.33-39 of text

(stopped before Arithmetic in Other Bases)

Solve in notebook as individual work for

next class:

1, 2, 3, 4, 5, 20, 21

22

QUIZ: Convert to decimal

1101 00112 =

AB716 =

5138 =

6928 = 23

Addition in Binary

24

Remember that there are only 2 digits in binary,

0 and 1

1 + 1 is 0 with a carry

Carry Values 0 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 QUIZ

25

Carry values

go here

1 0 1 0 1 1 0

+1 0 0 0 0 1 1

14

Check in base ten!

Subtraction in Binary

26

Remember borrowing?

1 2

0 2 0 2

1 0 1 0 1 1 1

- 1 1 1 0 1 1

0 0 1 1 1 0 0

15

Borrow values

Check in base ten!

Subtraction QUIZ

27

1 0 1 1 0 0 0

- 1 1 0 1 1 1

15

Check in base ten!

Borrow values

go here

Direct conversions between bases that are powers of 2

28

binary

hexadecimal

octal

Counting

29

QUIZ: Add the hex column!

30

Conclusion:

• In order to represent any octal digit, we need at most ______ bits

• In order to represent any hex digit, we need at most ______ bits

31

Converting Binary to Octal

32

• 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

33

• Mark groups of four (from right)

• Convert each group

10101011 1010 1011

A B

10101011 is AB in base 16

18

Converting Octal to Hexadecimal

34

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

Extra-credit QUIZ: Convert from hex to octal 2AF16 = 8

Show all your work for credit!

35

?

Today we’ve covered pp.39-42 of text

(stopped before Converting from Base 10 to

Other Bases)

Solve in notebook as individual work for

next class: 6 through 11

36

Homework for ch.1 is due at the beginning of next class (this Friday)

37

Subtraction QUIZ

38

1 0 1 0 0 0 0

- 1 0 0 1 0 1

15

Borrow values

go here

Check in base ten!

Periodic patterns

39

The inverse problem: Converting Decimal to Other Bases

40

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

answer Replace the original decimal number with the quotient

Algorithm for converting a number in base

10 to any other base R:

19

Known as repeated division (by the base)

Converting Decimal to Binary

41

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

LSB MSB

Repeated division QUIZ

42

Convert 4210 to binary

42 2 = rem.

4210 = 2

19

The repeated division algorithm can be used to convert from any

base into any other base (but normally we use it only for 10 → 2)

43

Read text example on p.43: Converting Decimal to Hex using repeated division

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

46

• Motivated by the 6-bit codes for printable graphic

patterns created by the U.S. Army and Navy

• 6, 18, 26, 36, 48-bit words

• Some history: – 18-Bit Computers from DEC

– 36-bit Wikipedia

• Edged out of the market by the need for floating-point numbers – IBM System/360 (1965)

– 8-bit microprocessors (1970s)

6-bit computers: an “Evolutionary dead-end”?

47

Unisys is still successful with their 36- and 48-bit machines :

• Clearpath Dorado line of 36-bit CISC high-end servers

• Clearpath Libra line of 48-bit mainframes

… although they are being transitioned to Intel Xeon chips (64-bit): see article

6-bit computers: an “Evolutionary dead-end”?

Grace Murray Hopper

48

• Ph.D. in mathematics

• Wrote “A-0”, the world’s first

compiler, in 1952!

• Co-invented COBOL

• Found the first “bug” in Mark II

• Rear-admiral of the US Navy

• “Nanosecond” wires

From the history of computing: bi-quinary

49 The front panel of the legendary

IBM 650 (source: Wikipedia)

Roman abacus (source: MathDaily.com)

Ethical Issues →Tenth Strand

50

What do the following acronyms stand for:

• ACM ?

• IEEE ? What is the tenth strand? Why the “tenth”?

51

Why the “tenth”? A: In the 1989 ACM report (“Computing as a Discipline” p.12), the following 9 areas (strands) of CS were defined:

• Algorithms and data structures • Programming languages • Computer Architecture • Numerical and symbolic computations • Operating systems (OS) • Software engineering • Databases • Artificial intelligence (AI) • Human-computer interaction

Ethical Issues →Tenth Strand

52

The latest official release of the IEEE/ACM CS Curriculum was in 2001:

• 3 levels of organization: areas → knowledge units → topics

• There are now 14 areas, and the “tenth strand” is listed in position 12

• What does “SP” stand for?

• How many knowledge units does the “SP” area have?

• Name two of these units!

Chapter Review questions

• Describe positional notation

• Convert numbers in other bases to base 10

• Convert base-10 numbers to numbers in other bases

• Add and subtract in binary

• Convert between bases 2, 8, and 16 using groups of digits

• Count in binary

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

53 6 24

Individual work for next class:

solve in notebook end-of-chapter exercises

22, 26

54

Homework Due next Friday, Sept. 14:

End-of-chapter ex. 23, 24, 25, 28, 29, 30, 33, 38

The latest homework assigned is always available on the course webpage

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