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Lecture 1, 23-July-20Discrete Mathematics 2003

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DISCRETE MATHEMATICS, 2003

Lecturers: Simon Smith (mostly)& Christopher Lenard. Simon

(Room Bus 2.12) is the subjectcoordinator.

Text:Discrete Mathematics forComputingby Peter Grossman secondedn

Assessment: two 45-min tests (each 20%) anda 3-hour exam (60%).

Tests: in lectures on Friday 29 August and 26September

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Tutorials and Web Support

Tutorials: There are 4 tute groups:

Mon 10am, Noon, 4pm & Wed Noon

You need to sign up ASAP for a tutorial

tutes start in Week 2

Web Page: www.bendigo.latrobe.edu.au/

mte/maths/staff/smith/discrete/

PDF versions of PowerPoint slides used in

lectures are available at this web site

(usually the day before the lecture)

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2.1 Real Numbers and the

Decimal Number System

Real Numbers are the familiar numbers ofeveryday life. Important types are:

Natural numbers 1, 2, 3, 4, 5, .Integers 0, 1, 1, 2, 2, 3, 3, .

Rational numbers can be written asm/n, where m,n are integers and n is not 0 e.g. 2/5, 13/721

Irrational numbers are the real nos thatarent rational

http://www.bendigo.latrobe.edu.au/mte/maths/staff/smith/discrete/http://www.bendigo.latrobe.edu.au/mte/maths/staff/smith/discrete/http://www.bendigo.latrobe.edu.au/mte/maths/staff/smith/discrete/http://www.bendigo.latrobe.edu.au/mte/maths/staff/smith/discrete/

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Lecture 1, 23-July-20Discrete Mathematics 2003

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Rational & Irrational Numbers

Every rational no. can be written as either a

terminating decimal (e.g. 1 = 1.75) or as a

recurring decimal (e.g. 2/3 = 0.666666.)

The irrational nos are the real nos whose

decimal expansions neither terminate nor

recur. Examples include:

2 = 1.41421356237309504880168872.

= 3.14159265358979323846264338.

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Place Value and Base

A number such as 6245.37 is in decimal

form, with each digit having aplace value

In decimal form, place values are powers of

10 so the decimal system is said to have a

base of 10

Note that base 10 requires ten digits (i.e.

09)

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2.2 The Binary Number System

Simplest number system is base 2, or binary

Uses the 2 digits (bits) 0 and 1

Used exclusively in computers (ON/OFFswitches, magnetised/unmagnetised

memory elements)

A typical binary number is 1011.1012

The subscript 2 denotes the base the base

should be included if it is not 10

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Converting Binary to Decimal

Example: Convert 1011.1012 to decimal Solution: 1011.1012

= (123) + (022) + (121) + (120)

+ (121) + (022) + (123)

= 8 + 2 + 1 + 0.5 + 0.125

= 11.625

Exercise: Convert 110001.0112 to decimal

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2.3 Conversion from Decimal to

Binary

Well begin by converting integers

Example: Convert 183 to binary

Solution: Note that the powers of 2 are1, 2, 4, 8, 16, 32, 64, 128, 256, 512, .

Now write 183 using just these powers.

Thus 183 = 128 + 55

= 128 + 32 + 23 = 128 + 32 + 16 + 7

= 128 + 32 + 16 + 4 + 2 + 1 = 101101112

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Decimal to Binary - A Better Way

Previous method is awkward for large nos.

A better method is to repeatedly divide by 2,writing down the quotientand remainderat

each step, until the quotient is zero. Now write down the remainders in reverse

order this is the binary form of the integer

Example: Convert 183 to binary

Exercise: Convert 212 to binary

Answer: 212 = 110101002