Enjoy Reading Books - Mathematics for Computer Scientists

Mathematics for Computer Scientists — read online for free the complete book (whole text) full work

Below is the text of the book, divided by pages. System saving the place of the last page read, allows you to conveniently read the book "Mathematics for Computer Scientists" online for free, without having to search again every time where you left off. Put a bookmark, and you can go to the page where you finished reading at any time.

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

1234567...40

Gareth J. Janacek & Mark Lemmon Close

Mathematics for Computer Scientists

Mathematics for Computer Scientists

2014 Gareth J. Janacek, Mark Lemmon Close & Ventus Publishing ApS ISBN 978-87-7681-426-7

Contents

Numbers

The statement calculus and logic

Mathematical Induction

Sets

Counting

Functions

Sequences

Calculus

Algebra: Matrices, Vectors etc.

Probability

Looking at Data

Introduction

The aim of this book is to present some the basic mathematics that is needed by computer scientists. The reader is not expected to be a mathematician and we hope willnd what follows useful.

Just a word of warning. Unless you are one of the irritating minority math- ematics is hard. You cannot just read a mathematics book like a novel. The combination of the compression made by the symbols used and the precision of the argument makes this impossible. It takes time and eort to decipher the mathematics and understand the meaning.

It is a little like programming, it takes time to understand a lot of code and you never understand how to write code by just reading a manual - you have to do it! Mathematics is exactly the same, you need to do it.

Chapter 1 Numbers

Defendit numerus: There is safety in numbers

We begin by talking about numbers. This may seen rather elementary but is does set the scene and introduce a lot of notation. In addition much of what follows is important in computing.

1.0.1 Integers

We begin by assuming you are familiar with th e integers

1,2,3,4 , .. . ,101,102 ,... , n,... , 2 32582657 1, .. . ,

sometime called the whole numbers. These are just the numbers we use for count- ing. To these integers we add th e zer o , 0, dened as

0+ any intege rn=0+n=n+0= n

Once we have the integers and zero mathematicians create negative integers by denin g(n) as:

the number which when added t on gives zero, s on + (n ) = (n ) +n= 0.

Eventually we get fed up with writin gn + (n ) =0 and write this a snn=0 .

We have now got the positive and negative integer s{... , 3 , 2 , 1, 0, 1, 2, 3, 4, .. . }

You are probably used to arithmetic with integers which follows simple rules.

To be on the safe side we itemize them, so for integer sa an d b 1 .a+b=b+ a

ab=ba o r a b= ba

ab= ab

4 .(a)(b ) = ab

5. To save space we writ e a k as a shorthand fo ra multiplied by itsel fk times. So3 =3333 an d = . Not e a n a m = a n + m

6. Do note tha t n =1.

Factors and Primes

Many integers are products of smaller integers, for exampl e 2 3 7= 4. Here

2, 3 and 7 are called th e factor s of 42 and the splitting of 42 into the individual components is known a s factorizatio n . This can be a dicult exercise for large

integers, indeed it is so dicult that it is the basis of some methods in cryptography.

Of course not all integers have factors and those that do not, such as

3, 5, 7, 11, 13,... , 216091 1,...

are known a s prime s . Primes have long fascinated mathematicians and others see

http://primes.utm.edu/,

and there is a considerable industry looking for primes and fast ways of factorizing integers.

To get much further we need to consider division, which for integers can be tricky since we may have a result which is not an integer. Division may give rise to a remainde r , for example

9=24+ 1.

and so if we try to divide 9 by 4 we have a remainder of 1 .

In general for any integer sa an d b

b=ka+ r

wher er is th e remainder . I fr is zero then we sa ya divide s b writte na|b . A single vertical bar is used to denot e divisibilit y . For exampl e2| 12 8,7| 4but 3 d o es not divide 4, sym b olicall y 3.

Aside

Tond the factors of an integer we can just attempt division by primes i.e.

2, 3, 5, 7, 11, 19, .. .. If it is divisible b yk the nk is a factor and we try again. When we cannot divide b yk we take the next prime and continue until we are left with a prime. So for example:

1. 2394/2=1197 cant divide by 2 again so try 3

2. 1197/3=399

3. 399/3 = 133 cant divide by 3 again so try 7 ( not divisible by 5) 4. 133/7 = 19 which is prime so 2394 =2337 19

Modular arithmetic

Th e mo d operator you meet in computer languages simply gives the remainder after division. For example,

1 . 2mo d4=1 becaus e 2 54=6 remainder 1. 2 . 1mo d5=4 sinc e 1 9=35+4 .

3 . 2mo d5=4 .

4 . 9mo d 1 1=0 .

There are some complications when negative numbers are used, but we will ignore them. We also point out that you will often see these results written in a slightly dierent way i.e . 2 4=4 mo d5 o r 2 1=0 mo d7 . which just mean s 2mo d5 =

an d 21 o d7=

Modular arithmetic is sometimes called clock arithmetic. Suppose we take a

24 hour clock so 9 in the morning is 09.00 and 9 in the evening is 21.00. If I start a journey at 07.00 and it takes 25 hours then I will arrive at 08.00. We can think of this as 7+25 = 32 and 32 mod 24 = 8. All we are doing is starting at 7 and going around the (25 hour) clock face until we get to 8. I have always thought this is a complex example so take a simpler version.

Four people sit around a table and we label their positions 1 to 4. We have a

pointer point to position 1 which we spin. Suppose it spins 11 and three quarters or 47 quarters. The it is pointing a t 4mo d4 or 3.

The Euclidean algorithm

Algorithms which are schemes for computing and we cannot resist putting one in at this point. The Euclidean algorithm fornding the gcd is one of the oldest algorithms known, it appeared in Euclids Elements around 300 BC. It gives a way ofnding the greatest common divisor (gcd) of two numbers. That is the largest number which will divide them both.

Our aim is tond a a way ofnding the greatest common divisor, gc d( a, b) of two integer sa an db .

Suppos ea is an integer smaller tha nb .

1. Then tond the greatest common factor betwee na an db , divid eb b ya . If the remainder is zero, the nb is a multiple o fa and we are done.

Next page

1234567...40

Light

Font size:

↓

↑

Georgia Tahoma Arial Verdana

Reset

Interval:

↓

↑

Bookmark:

Make

Similar books «Mathematics for Computer Scientists»

Look at similar books to Mathematics for Computer Scientists. We have selected literature similar in name and meaning in the hope of providing readers with more options to find new, interesting, not yet read works.

Mangey Ram

Recent Advances in Mathematics for Engineering

edulink GmbH

Practice Test for the TestAS Mathematics, Computer Science and Natural Sciences (Preparation Book for the TestAS Mathematics, Computer Science and Natural Sciences 3)

Ronald Graham

Concrete Mathematics: A Foundation for Computer Science

Jon Pierre Fortney

Discrete Mathematics for Computer Science: An Example-Based Introduction

Klaus Mainzer

The Digital and the Real World: Computational Foundations of Mathematics, Science, Technology, and Philosophy

Mark Lemmon Close

Mathematics for Computer Scientists

John Vince

Mathematics for Computer Graphics

Bagdasar

Concise Computer Mathematics: Tutorials on Theory and Problems

Larry Wasserman

All of Statistics

Daniel J. Velleman

How to Prove It: A Structured Approach

Paul DuChateau

Advanced Mathematics for Engineers and Scientists

Charles F. Van Loan

Introduction to computational science and mathematics

Discussion, reviews of the book Mathematics for Computer Scientists and just readers' own opinions. Leave your comments, write what you think about the work, its meaning or the main characters. Specify what exactly you liked and what you didn't like, and why you think so.