Lecture 1

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Number SystemsNumber Systems

Hexadecimal

Decimal Octal

Binary

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Lecture Template:Lecture Template:

Types of number systemsTypes of number systems Number basesNumber bases Range of possible numbersRange of possible numbers Conversion between number basesConversion between number bases Common powersCommon powers Arithmetic in different number basesArithmetic in different number bases Shifting a numberShifting a number

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Types of Number SystemsTypes of Number Systems

Additive:Numbers have intrinsic value: e.g.: Roman Numerals: LVIII = 50 + 5 + 1 + 1 + 1 = 58

Positional: Value depends on position: e.g.: Decimal system: 55 = 5 x 10 + 5 x 1

Additive Number Systems are not used much any more:

Awkward to use.Prone to errors.

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DefinitionsDefinitions

The Base of a number system – how many different digits (incl. zero) are used in the system.

Base 2: 0, 1

Base 5: 0, 1, 2, 3, 4

Base 8: 0, 1, 2, 3, 4, 5, 6, 7

Base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Base 16: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

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DefinitionsDefinitions

Bit – a cell holding a single binary number (0 or 1)

Byte = 8 bits (can hold 28 = 256 different patterns/values)

Word – a fixed-sized group of bits that the computer handles together. Typical word sizes: 4, 8, 16, 32, 64, 128 bits

1K = 1024 bytes

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Magnetic Core MemoryMagnetic Core Memory

http://en.wikipedia.org/wiki/Computer

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Common Number SystemsCommon Number Systems

System Base SymbolsUsed by humans?

Used in computers?

Decimal 10 0, 1, … 9 Yes No

Binary 2 0, 1 No Yes

Octal 8 0, 1, … 7 No Yes

Hexa-decimal

16 0, 1, … 9,A, B, … F

No Yes

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Positional decimal systemPositional decimal system

The number 125 means:

1 group of 100 (100 = 102) 2 groups of 10 (10 = 101) 5 groups of 1 (1 = 100)

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Place values (1 of 2)Place values (1 of 2)

In our usual positional number system, the meaning of a digit depends on where it is located in the number

3 groups of 1000

7 groups of 100

3 groups of 10

2 groups of 1

Example: 3 7 3 2

10

Place values (2 of 2)Place values (2 of 2)

12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100

125

Base

Weight

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Representing in bases: Representing in bases: 1010, , 22, , 88, , 1616

86510 = 8 x 102 + 6 x 101 + 5 x 100 = 800 + 60 + 5

10112 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 8 + 2 + 1 = 1110

258 = 2 x 81 + 5 x 80 = 16 + 5 = 2110 A716 = 10 x 161 + 7 x 160 = 160 + 7 =

16710

Note: The subscript naming the base is itself given in base ten (10), by convention.

Base

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Counting in bases (1 of 3)Counting in bases (1 of 3)

Decimal Binary OctalHexa-

decimal

0 0 0 0

1 1 1 1

2 10 2 2

3 11 3 3

4 100 4 4

5 101 5 5

6 110 6 6

7 111 7 7

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decimal

8 1000 10 8

9 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F

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decimal

16 10000 20 10

17 10001 21 11

18 10010 22 12

19 10011 23 13

20 10100 24 14

21 10101 25 15

22 10110 26 16

23 10111 27 17

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Estimating magnitude: BinaryEstimating magnitude: Binary

1101 01102 = 21410

1101 01102 > 19210 (128 + 64 + additional bits to the right)

Place 27 26 25 24 23 22 21 20

Value 128 64 32 16 8 4 2 1

Evaluate 1 x 128 1 x 64 0 x 32 1 x16 0 x 8 1 x 4 1 x 2 0 x 1

Sum for Base 10

128 64 0 16 0 4 2 0

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Range of possible numbersRange of possible numbers

R = BK where R = rangeB = baseK = number of digits

Example #1: Base 10, 2 digitsR = 102 = 100 different numbers (0…99)

Example #2: Base 2, 16 digitsR = 216 = 65,536 or 64K16-bit PC can store 65,536 different number values

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Decimal Range for Bit WidthsDecimal Range for Bit Widths

Bits Digits Range

1 0+ 2 (0 and 1)

4 1+ 16 (0 to 15)

8 2+ 256

10 3 1,024 (1K)

16 4+ 65,536 (64K)

20 6 1,048,576 (1M)

32 9+ 4,294,967,296 (4G)

64 19+ Approx. 1.6 x 1019

128 38+ Approx. 2.6 x 1038

See also Ch. 3, p. 74

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Conversion Among BasesConversion Among Bases

Hexadecimal

Decimal Octal

Binary

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Binary to Decimal (1 of 3)Binary to Decimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueMultiply each bit by 2n, where n is the “weight” of the bitThe weight is the position of the bit, starting from 0 on the rightAdd the results

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1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32

4310

Bit “0”

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Octal to Decimal (1 of 3)Octal to Decimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueMultiply each bit by 8n, where n is the “weight” of the bitThe weight is the position of the bit, starting from 0 on the rightAdd the results together

Note: 80 = 1, 81 = 8, 82 = 64 83 = 512, Etc.

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7248 => 4 x 80 = 4 x 1 = 42 x 81 = 2 x 8 = 167 x 82 = 7 x 64 = 448

46810

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Hexadecimal to Decimal (1 of 3)Hexadecimal to Decimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

26


TechniqueMultiply each bit by 16n, where n is the “weight” of the bitThe weight is the position of the bit, starting from 0 on the rightAdd the results

Note: 160 = 1, 161 = 16, 162 = 256 163 = 4096, Etc.

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ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560

274810

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Decimal to Binary (1 of 3)Decimal to Binary (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueDivide by two, keep track of the remainderFirst remainder is bit 0 (LSB, least-significant bit)Second remainder is bit 1Etc.

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12510 = ?2 2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1

12510 = 11111012

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Octal to Binary (1 of 3)Octal to Binary (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueConvert each octal digit to a 3-bit equivalent binary representation

See Table, Slides 12-14

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7058 = ?2

7 0 5

111 000 101

7058 = 1110001012

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Hexadecimal to Binary (1 of 3)Hexadecimal to Binary (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueConvert each hexadecimal digit to a 4-bit equivalent binary representation


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10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112

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Decimal to Octal (1 of 3)Decimal to Octal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueDivide by 8Keep track of the remainder

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123410 = ?8

8 1234 154 28 19 28 2 38 0 2

123410 = 23228

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Decimal to Hexadecimal (1 of 3)Decimal to Hexadecimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueDivide by 16Keep track of the remainder

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123410 = ?16

123410 = 4D216

16 1234 77 216 4 13 = D16 0 4

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Binary to Octal (1 of 3)Binary to Octal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueGroup bits in threes, starting on rightConvert to octal digits


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10110101112 = ?8

1 011 010 111

1 3 2 7

10110101112 = 13278

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Binary to Hexadecimal (1 of 3)Binary to Hexadecimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueGroup bits in fours, starting on rightConvert to hexadecimal digits


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10101110112 = ?16

10 1011 1011

2 B B

10101110112 = 2BB16

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Octal to Hexadecimal (1 of 3)Octal to Hexadecimal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueUse binary as an intermediary


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10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16

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Hexadecimal to Octal (1 of 3)Hexadecimal to Octal (1 of 3)

Hexadecimal

Decimal Octal

Binary

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TechniqueUse binary as an intermediary


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1F0C16 = ?8

1 F 0 C

0001 1111 0000 1100

1 7 4 1 4

1F0C16 = 174148

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Exercise – Convert ...Exercise – Convert ...

Don’t use a calculator!


decimal

33

1110101

703

1AF

Skip answer Answer

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Exercise – Convert (answers)Exercise – Convert (answers)


decimal

33 100001 41 21

117 1110101 165 75

451 111000011 703 1C3

431 110101111 657 1AF

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Common Powers (1 of 2)Common Powers (1 of 2)

Base 10Power Preface Symbol

10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

103 kilo k

106 mega M

109 giga G

1012 tera T

Value

.000000000001

.000000001

.000001

.001

1000

1000000

1000000000

1000000000000

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Common Powers (2 of 2)Common Powers (2 of 2)

Base 2 Power Preface Symbol

210 kilo k

220 mega M

230 Giga G

Value

1024

1048576

1073741824

What is the value of “k”, “M”, and “G”?In computing, particularly w.r.t. memory,

the base-2 interpretation generally applies

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ExampleExample

/ 230 =

1. Double click on My Computer2. Right click on C:3. Click on Properties

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Multiplying powersMultiplying powers

For common bases, add powers

26 210 = 216 = 65,536

or…

26 210 = 64 210 = 64k

ab ac = ab+c

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Table of powersTable of powers

PowerBase 8 7 6 5 4 3 2 1 0

2 256 128 64 32 16 8 4 2 1

8 32,768 4,096 512 64 8 1

16 65,536 4,096 256 16 1

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FractionsFractions

Number point or radix pointDecimal point in base 10Binary point in base 2

No exact relationship between fractional numbers in different number bases

Exact conversion may be impossible

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Decimal fractionsDecimal fractions

Move the number point one place to the right

Effect: multiplies the number by the base numberExample: 139.010 139010

Move the number point one place to the left

Effect: divides the number by the base numberExample: 139.010 13.910

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Fractions: Base Fractions: Base 1010

Place 10-1 10-2 10-3 10-4

Value 1/10 1/100 1/1000 1/10000

Evaluate 2 x 1/10 5 x 1/100 8 x 1/1000 9 x1/10000

Sum .2 .05 .008 .0009

.258910

10-1 10-2 10-3 10-4

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Fractions: Base Fractions: Base 22

.1010112 = 0.67187510

Place 2-1 2-2 2-3 2-4 2-5 2-6

Value 1/2 1/4 1/8 1/16 1/32 1/64

Evaluate 1 x 1/2 0 x 1/4 1x 1/8 0 x 1/16 1 x 1/32 1 x 1/64

Sum .5 0.125 0.03125 0.015625

2-1 2-2 2-3 2-4 2-5 2-6

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Fractions: Base 10 and Base 2Fractions: Base 10 and Base 2

No general relationship between fractions of types 1/10k and 1/2k

Therefore a number representable in base 10 may not be representable in base 2But: the converse is true: all fractions of the form 1/2k can be represented in base 10

Fractional conversions from one base to another are stopped

If there is a rational solution, orWhen the desired accuracy is attained

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Fractions: From Base Fractions: From Base BB To Base 10To Base 10

Determine the appropriate weight for each fractional digit as a negative power of the base

Multiply each digit by its weight Add the values Example:

.A116 = 10x16-1 + 1x16-2 = 10x0.0625 + 1x0.00390625 = 0.6289062510

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Fractions: From Base 10 To Base Fractions: From Base 10 To Base BB

Multiply the fraction by the base value B B (i.e., 2, 8 or 16)

Record the values that move to the left of the radix point and drop them

Repeat the process until the value being multiplied is zero, orthe desired number of digits of accuracy is attained

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Fractions: From Base 10 To Base Fractions: From Base 10 To Base BB

0.67312510 = ?2 0.673125 x 2 1.346250 x 2 0.692500 x 2 1.385000 x 2 0.770000 x 20.1010112 1.540000 x 2 1.080000

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Fractions: From Base 2 To Base Fractions: From Base 2 To Base 8, 168, 16

Group digits from left to right in groups of 3 (base 8) or 4 (base 16)

Supplement the right-most group with 0’s, if necessary

Convert each group to the desired base.


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Fractions: From Base Fractions: From Base 8, 168, 16 To Base 2 To Base 2

Convert each octal (base 8) or hexadecimal (base 16) digit to its 3-bit or 4-bit representation


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Fractions: between Base Fractions: between Base 8 8 andand Base Base 1616

Use binary conversion as an intermediaryintermediary

Example:0.C816 = ?8

1100 10002

6 2 0 0.C816 = 0.628

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Mixed number conversionMixed number conversion

Convert whole part and fraction part separately


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Arithmetic operations (1 of 14)Arithmetic operations (1 of 14)

Binary Addition Two 1-bit values

A B A + B

0 0 0

0 1 1

1 0 1

1 1 100 and carry 1 to the

next more significant bit, i.e.

“two”

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Two n-bit valuesAdd individual bits (see Table)Propagate carries

10101 21+ 11001 + 25 101110 46

11

Note: superscripts are carried amounts.

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Addition (different bases) (3 of 14)Addition (different bases) (3 of 14)

Base Problem Largest Single Digit

Decimal 6

+3 9

Octal 6+1 7

Hexadecimal 6

+9 F

Binary 1+0 1

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Addition (different bases) (4 of 14)Addition (different bases) (4 of 14)

Base Problem Carry Answer

Decimal 6

+4 Carry the 10 10

Octal 6+2 Carry the 8 10

Hexadecimal 6

+A Carry the 16 10

Binary 1+1 Carry the 2 10

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Addition Table: Base 10 (5 of 14)Addition Table: Base 10 (5 of 14)

+ 0 1 2 3 4 5 6 7 8 9

0 0 1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

4 4 5 6 7 8 etc

9 10 11 12 13

310 + 610 = 910

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Addition Table: Base 8 (6 of 14)Addition Table: Base 8 (6 of 14)

+ 0 1 2 3 4 5 6 7

0 0 1 2 3 4 5 6 7

1 1 2 3 4 5 6 7 10

2 2 3 4 5 6 7 10 11

3 3 4 5 6 7 10 11 12

4 4 5 6 7 10 11 12 13

5 5 6 7 10 11 12 13 14

6 6 7 10 11 12 13 14 15

7 7 10 11 12 13 14 15 16

38 + 68 = 118

80


Binary Subtraction Two 1-bit values

A B A - B

0 0 0

0 1 1

1 0 1

1 1 0

and borrow 1 from the next more significant bit

81


Two n-bit valuesSubtract individual bits (see Table)Keep track of the borrowings

00100101 – 00010001 = 00010100

00100101 3710

- 00010001 - 1710

00010100 2010

0 10

82


MultiplicationDecimal (just for fun)

35x 105 175 000 35 3675

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Binary multiplication, two 1-bit values

A B A B

0 0 0

0 1 0

1 0 0

1 1 1

84


Binary multiplication, two n-bit valuesAs with decimal values

1110 x 1011 1110 1110 0000 111010011010

85

Multiplication Table: Base 10 (12 Multiplication Table: Base 10 (12 of 14)of 14)

x 0 1 2 3 4 5 6 7 8 9

0 0

1 1 2 3 4 5 6 7 8 9

2 2 4 6 8 10 12 14 16 18

3 3 6 9 12 15 18 21 24 27

4 0 4 8 12 16 20 24 28 32 36

5 5 10 15 20 25 30 35 40 45

6 6 12 18 24 30 36 42 48 54

7 7 14 21 28 35 42 49 56 63

etc.

310 x 610 = 1810

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Multiplication Table: Base 8 (13 of Multiplication Table: Base 8 (13 of 14)14)

x 0 1 2 3 4 5 6 7

0 0

1 1 2 3 4 5 6 7

2 2 4 6 10 12 14 16

3 0 3 6 11 14 17 22 25

4 4 10 14 20 24 30 34

5 5 12 17 24 31 36 43

6 6 14 22 30 36 44 52

7 7 16 25 34 43 52 61

38 x 68 = 228

87


Binary division, two n-bit valuesAs with decimal values

100001/11 = 1011 11 ) 100001

1011

11

0010011

1111

0

divisordividend

quotient

88

Shifting a numberShifting a number

Shifting a decimal number to the left by one position is equivalent to multiplying by 10

Shifting a binary number to the left by one position is equivalent to multiplying by 2

General rule: shifting a number in any base left one digit multiplies its value by the base; shifting one digit right divides its value by the base

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Thank you!Thank you!Reading: Lecture slides and notes, Chapter 3Reading: Lecture slides and notes, Chapter 3

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Lecture 1