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John Magee 15 July 2013

Some material copyright Jones and Bartlett

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!!! MATH WARNING !!! TODAY’S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS WILL BE ALLOWED ON THE QUIZ (and that you probably won’t need them)

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Overview/Questions – What gives a number its value? – What is a number system? – I’ve heard that computers use binary numbers. What’s a binary number? – What kind of numbers do computers store and manipulate?

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Numbers Natural Numbers 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 6

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Numbers 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

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Numbering Systems A numbering system assigns meaning to the position of the numeric symbols. For example, consider this set of symbols: 642 What number is it? Why?

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Numbering Systems It depends on the numbering system.

642 is 600 + 40 + 2 in BASE 10 The base of a number determines the number of digits (e.g. symbols) and the value of digit positions

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Positional Notation 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 number

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Positional Notation 642 is 63 * 102 + 42 * 101 + 21 * 100 B is the base of the number

As a general form:

dn * Bn-1 + dn-1 * Bn-2 + ... + d1 * B0

n is the number of digits in the number

d is the digit in the ith position in the number 11

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What Would Pooh Do?

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Binary Numbers Digital computers are made up of electronic circuits, which have exactly 2 states: on and off.

Computers use a numbering system which has exactly 2 symbols, representing on and off.

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Binary Numbers 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, so we use only 2 symbols: 0,1

For a given base, valid numbers will only contain the digits in that base, which range from 0 up to (but not including) the base.

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Counting… Let’s remember Kindergarten…

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Binary Numbers and Computers A binary digit or bit can take on only these two values. Low Voltage = 0 High Voltage = 1

all bits have 0 or 1

Binary numbers are built by concatenating a string of bits together. Example: 10101010 16

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Positional Notation: Binary Numbers Recall this general form: dn * Bn-1 + dn-1 * Bn-2 + ... + d1 * B0

The same can be applied to base-2 numbers: 1011(binary) =1 * 23 (1 * 8) + 0 * 22 (0 * 4) + 1 * 21 (1 * 2) + 1 * 20 (1 * 1) 1011(binary) =8 + 0 + 2 + 1 = 11(decimal) 17

Converting Binary to Decimal What is the decimal equivalent of the binary number 01101110? (you try it! Work left-to-right)

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Converting Binary to Decimal What is the decimal equivalent of the binary number 01101110? 0 x 27 + 1 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 1 x 21 + 0 x 2º

= = = = = = = =

0 x 128 = 0 1 x 64 = 64 1 x 32 = 32 0 x 16 = 0 1x8 =8 1x4 =4 1x2 =2 0x1 =0 = 110 (decimal) 19

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Converting Binary to Decimal Try another one. What is the decimal equivalent of the binary number 10101010? (you try it! Work left-to-right)

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Converting Binary to Decimal Try another one. What is the decimal equivalent of the binary number 10101010? 1 x 27 + 0 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 0 x 2º

= = = = = = = =

1 x 128 = 128 0 x 64 = 0 1 x 32 = 32 0 x 16 = 0 1x8 =8 0x4 =0 1x2 =2 0x1 =0 = 170 (decimal) 21

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Converting from Decimal to Other Bases Algorithm (process) for converting number in base 10 to other bases While (the quotient is not zero) Divide the decimal number by the new base* Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient

* Using whole number (integer) division only. Example: 3 / 2 gives us a quotient of 1 and a remainder 1

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Converting Decimal to Binary What is the binary equivalent of the decimal number 103? 103 / 2 = 51, remainder 1 rightmost bit 51 / 2 = 25, remainder 1 25 / 2 = 12, remainder 1 12 / 2 = 6, remainder 0 6 / 2 = 3, remainder 0 3 / 2 = 1, remainder 1 1 / 2 = 0, remainder 1 leftmost bit 103 (decimal) = 1 1 0 0 1 1 1 (binary) 23

Converting Decimal to Binary Now you try one. What is the binary equivalent of the decimal number 201? Recall the algorithm: While (the quotient is not zero) Divide the decimal number by the new base* Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient

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Converting Decimal to Binary What is the binary equivalent of the decimal number 201? 201 / 2 = 100, remainder 1 rightmost bit 100 / 2 = 50, remainder 0 50 / 2 = 25, remainder 0 25 / 2 = 12, remainder 1 12 / 2 = 6, remainder 0 6 / 2 = 3, remainder 0 3 / 2 = 1, remainder 1 1 / 2 = 0, remainder 1 leftmost bit 201 (decimal) = 1 1 0 0 1 0 0 1 (binary) 25

Binary and Computers Byte 8 bits – a common unit of computer memory. Word A computer word is a group of bits which are passed around together during computation. The word length of the computer’s processor is how many bits are grouped together. • • • •

8-bit machine (e.g. Nintendo Gameboy, 1989) 16-bit machine (e.g. Sega Genesis, 1989) 32-bit machines (e.g. Sony PlayStation, 1994) 64-bit machines (e.g. Nintendo 64, 1996) 26

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Common Number Systems Binary – base 2, has 2 symbols: 0,1 Octal – base 8, has 8 symbols: 0,1,2,3,4,5,6,7 Decimal – base 10, has 10 symbols: 0,1,2,3,4,5,6,7,8,9 Hexadecimal - base 16 has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F 27

Why Hexadecimal? Base 16 is a multiple of Base 2: 16 = 24

Each four bits map to a hex digit. Converts easily to and from binary 28

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Binary, Hexadecimal, Decimal Each four bits map to a hex digit. Hexadecimal prefix 0x???? – No inherent value, just means “treat as a hex number”

0x94D3 29

Hexadecimal to Decimal Convert each hex digit into 4 bits. Convert binary to decimal. Example: 0x94D3 = 1001 0100 1101 0011 = 215 + 212 + 210 + 27 + 26 + 24 + 21 + 20 = 32768 + 4096 + 1024 + 128 + 64 + 16 + 2 + 1 = 38099 (decimal)

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Conversions Between Number Systems Try some! http://www.mathsisfun.com/binary-decimalhexadecimal-converter.html

My phone number: 0x16FF8BA69 (or: 101101111111110001011101001101001)

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Take-Away Points – Symbols represent values – Number systems – Binary – Hexadecimal

– When do computers use decimal, octal, and hexadecimal numbers?

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