CSE1301 Computer Programming Lecture 32: Number Representation

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CSE1301Computer Programming

Lecture 32:Number Representation

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Topics

• Representing characters and integers• Converting binary to decimal• Converting decimal to binary• Adding binary numbers• Limitations

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Representing Characters

• ASCII encoding (American Standard Code for Information Interchange)

• Each character represented by a one-byte (8-bit) number

• Other representations possible too:– EBCDIC (used by older IBM machines)– Unicode (16-bit character codes, provides

enough codes for characters from all modern languages)

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Representing Integers

• We represent integers using a base-10 positional notation.

• So the number 80710 means:8104 + 0103 + 7102 + 1101 + 0100

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Representing Integers

• Computers represent integers using a base-2 positional notation.

• So the number 101011 means:

125 + 024 + 123 + 022 + 121 + 120

132 + 016 + 18 + 04 + 12 + 11

= 43 in decimal

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Representing Integers

• Curiously, ye Olde Englishe pub-keepers used the same system:

• So the number 101011 means:

1gallon + 0pottle + 1quart + 0pint + 1chopin + 1gill

14.6l + 02.3l + 11.1l + 00.6l + 10.3l + 10.1l

= 6.1 litres

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Representing Integers• The first few binary numbers are:

0000....................00001....................10010....................20011....................30100....................40101....................50110....................60111....................71000....................81001....................91010....................101011....................111100....................121101....................131110....................141111....................15

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Converting binary to decimal

Example: Convert the unsigned binary number 10011010 to decimal.

01011001

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Converting binary to decimal

01011001

2021222324252627

01234567

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Converting binary to decimal

0101100101234567

64 12481632128

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Converting binary to decimal

0101100101234567

64 12481632128

128 + 16 + 8 + 2 = 154 (or 27 + 24 + 23 + 21)

So, 10011010 in unsigned binary is 154 in decimal.

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Converting Decimal to/from Binary

Analogy: Giving a $69.10 change from largest to smallest denomination:

69.10 = 1 $50 bill + 19.10

19.10 = 1 $10 bill + 9.10

9.10 = 1 $5 bill + 4.10

4.10 = 2 $2 coin + 0.10

0.10 = 1 10c coin

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25 20212223242627

01234567

Converting Decimal to/from Binary

Bit position.

Decimal value.

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1248163264128

01234567

Converting Decimal to/from Binary

Bit position.

Decimal value.

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Decimal to Binary

Example: Convert the decimal number 105 to unsigned binary.

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Q. What is the highest power of 2 that is less than or equal to 105?

A. 64

164 12481632

0123456

Next, consider the difference: 105 - 64 = 41

128

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Q. What is the highest power of 2 that is less than or equal to 41?

A. 32

11248163264

0123456

Next, consider the difference: 41 - 32 = 9

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Q. What is the highest power of 2 that is less than or equal to 9?

A. 8

1 11248163264

0123456

Next, consider the difference: 9 - 8 = 1

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Q. What is the highest power of 2 that is less than or equal to 1?

A. 1

11 11248163264

0123456

Next, consider the difference: 1 - 1 = 0 Done!

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1

Finally, fill the empty boxes with zeros.

1001011248163264

0123456

So, 105 decimal is 1101001 in unsigned binary.

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Converting decimal to binary

• We've seen that you convert binary to decimal by multiplying and adding.

• So it's no surprise that you convert decimal to binary by dividing and subtracting.

• You can also convert decimal to binary by dividing and noting the remainder.

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Converting decimal to binary123 ÷2

61 remainder 1 ÷2

30 remainder 1 ÷2

15 remainder 0 ÷2

7 remainder 1 ÷2

3 remainder 1 ÷2

1 remainder 1 ÷2

0 remainder 1

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Converting decimal to binary123 ÷2

61 remainder 1 ÷2

30 remainder 1 ÷2

15 remainder 0 ÷2

7 remainder 1 ÷2

3 remainder 1 ÷2

1 remainder 1 ÷2

0 remainder 1

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Converting decimal to binary123 1111011 ÷261 remainder 1 ÷230 remainder 1 ÷215 remainder 0 ÷27 remainder 1 ÷23 remainder 1 ÷21 remainder 1 ÷20 remainder 1

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Converting decimal to binary102 ÷2

51 remainder 0 ÷2

25 remainder 1 ÷2

12 remainder 1 ÷2

6 remainder 0 ÷2

3 remainder 0 ÷2

1 remainder 1 ÷2

0 remainder 1

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Converting decimal to binary102 ÷2

51 remainder 0 ÷2

25 remainder 1 ÷2

12 remainder 1 ÷2

6 remainder 0 ÷2

3 remainder 0 ÷2

1 remainder 1 ÷2

0 remainder 1

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102 1100110 ÷2

51 remainder 0 ÷2

25 remainder 1 ÷2

12 remainder 1 ÷2

6 remainder 0 ÷2

3 remainder 0 ÷2

1 remainder 1 ÷2

0 remainder 1

Converting decimal to binary

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102 1100110 ÷251 remainder 0 ÷225 remainder 1 ÷212 remainder 1 ÷26 remainder 0 ÷23 remainder 0 ÷21 remainder 1 ÷20 remainder 1

Converting decimal to binary

Most significant

bit

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102 1100110 ÷251 remainder 0 ÷225 remainder 1 ÷212 remainder 1 ÷26 remainder 0 ÷23 remainder 0 ÷21 remainder 1 ÷20 remainder 1

Converting decimal to binary

Least significant

bit

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Adding binary numbers• For individual bits there are only four

possibilities:

0 0 1 1 + 0 + 1 + 0 + 1

0 1 1 10

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos:

11100101+ 1011101

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos.

1 11100101

+ 10111010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos: 1

11100101+ 1011101

10

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos:

1 1 11100101

+ 1011101010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos: 1 1 1

11100101+ 1011101

0010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos: 1 1 1 1

11100101+ 1011101

00010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos:

1 1 1 1 1 11100101

+ 1011101000010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos: 1 1 1 1 1 1

11100101+ 1011101

1000010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos: 1 1 1 1 1 1 1

11100101+ 1011101101000010

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Adding binary numbers• For multiple digits we do the same as in

base-10: we add corresponding bits and carry the twos:

11100101 229

+ 1011101 + 93101000010 322

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Adding binary numbers• But that only works for unsigned numbers.

(positive numbers)

• For signed numbers it's much more complicated.

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Limitations of representations• None of these number representations acts

exactly like the integers.

• Infinitely many integers, but only 2N binary numbers for a specific N-bit representation.

• That means some operations will overflow.

• If the program doesn't crash when that happens, it will simply produce wrong answers (which is worse!)

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Limitations of representations• Computer arithmetic is only an

approximation of integer arithmetic.

• Many of the arithmetic laws you're used to don't always hold. (See Tutorial 11 - Advanced Exercise for examples.)

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Reading• Brookshear: Sections 1.4 to 1.7

• D&D: Appendix E