mathematics

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MATH 10021 Core Mathematics I

undo each other recur over and over again in mathematics To do old-fashioned subtraction of large numbers without a calculator we will again use a tower method and again we will make sure to align the place-values of the two numbers Example 4 Subtract 1978¡322 Solution 4 Scratch work: 1 9 7 8 ¡ 3 2 2 1 6 5 6 So: 1978¡322 = 1656

PDF	The Project Gutenberg eBook ꉠ: An Introduction to Dec 6 2012 · Project Gutenberg’s An Introduction to Mathematics by Alfred North Whitehead This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever

What is fundamentals of mathematics?
Fundamentals of Mathematics is a work text that covers the traditional study in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who: wish to meet the prerequisites of higher level courses such as elementary algebra
What is modern mathematics?
Modern mathematics will only admit statements and de nitions and arguments which exclusively employ the few simple ideas about number and magnitude and variables on which the science is founded.
What are the three concepts of mathematics?
These three notions, of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science.

Numbers and Cryptography

Important Note The material in the Notes corresponds to and often ex-tends that in The Heart of Mathematics [HM]. See also the comments on page iv. The corresponding page numbers in [HM] are noted here in the mar-gin. First study the material in [HM], then study the more concentrated and extended treatment here. Additional material beyond that in [

John Hutchinson

(suggestions and comments to: John.Hutchinson@anu.edu.au) maths.anu.edu.au

What is Mathematics?

Mathematics is the study of pattern and structure. Mathematics is funda-mental to the physical and biological sciences, engineering and information technology, to economics and increasingly to the social sciences. The patterns and structures we study in mathematics are universal. It is perhaps possible to imagine a universe in which the biology and

These Notes and The Heart of Mathematics

[HM] is an excellent book. It is one of a small number of texts intended to give you, the reader, a feeling for the theory and applications of contemporary mathematics at an early stage in your mathematical studies. However, [HM] is directed at a di erent group of students undergraduate students in the United States with little mathematics backgr

Studying Mathematics

This takes time and e ort but it is very interesting material and intellectually rewarding. Do lots of Questions from [HM] and from these Notes, answer the questions here marked with a -and keep your solutions and comments in a folder. Material marked ? is not in [HM] and is more advanced. Some is a little more advanced and some is a lot more advan

Acknowledgements

I would like to thank Richard Brent, Tim Brook, Clare Byrne, Jonathan Manton, Neil Montgomery, Phoebe Moore, Simon Olivero, Raiph McPherson, Jeremy Reading, Bob Scealy, Lisa Walker and Chris Wetherell, for comments and suggestions on various drafts of these notes. maths.anu.edu.au

Quotations

Philosophy is written in this grand bookI mean the universe which stands continually open to our gaze, but it cannot be understood unless one rst learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other mathematical gur

Alfred North Whitehead Science and the Modern World [1925]

All the pictures which science now draws of nature and which alone seem capable of according with observational facts are mathematical pictures . . . . From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. maths.anu.edu.au

Sir James Hopwood Jeans The Mysterious Universe [1930]

1Simeon Poisson was the thesis adviser of the thesis adviser of

Characterising Irregularity, Science 200 [1978]

Mathematics is like a ight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process for sharpening our perception so that we discover patterns that are everywhere around. : : : To share in the delight and the intellectual experie

2.1 Counting2

Using estimation to move from qualitative to quantitative thinking and reasoning is a powerful tool. maths.anu.edu.au

The Pigeon Hole Principle

The following simple result has interesting and often surprising conclusions. maths.anu.edu.au

?The Principle of Mathematical Induction7

This is only discussed in [HM] in a very light way, to show that \\all numbers are interesting". Here we discuss and give some more serious examples. maths.anu.edu.au

2.2 The Fibonacci Sequence

Looking at simple things deeply, nding a pattern, and using the pattern to gain new insights provides great value. maths.anu.edu.au

Overview

In the remainder of this Chapter we will often say \ umber" when we mean an integer rather than a general real number. We do this to be consistent with [HM]. It should be clear from the context what we mean. maths.anu.edu.au

Sequences of Numbers

We usually write an (in nite) sequence of numbers in the form maths.anu.edu.au

?Formula for the nth Fibonacci Number

There is a formula for the nth Fibonacci number. This is tricky and is not done in [HM]. maths.anu.edu.au

2.3 Prime Numbers

Examining the building blocks of a complex structure answers old questions, invites new questions, and leads to greater understanding. maths.anu.edu.au

How Dense are the Primes?

Numerical Experimentation It is conventional to let maths.anu.edu.au

k 1 qi.

If we keep cancelling in this way we see that the number of factors on each side are the same and each factor occurs the same number of times on each side. This means that both factorisations are the same except possibly for a reordering of the factors. maths.anu.edu.au

2.4 Modular Arithmetic

Generalising a simple idea like telling time on a clock can lead to important applications. maths.anu.edu.au

2.5 RSA Public Key Cryptography

Things that seem abstract and devoid of application today may be central in our daily lives tomorrow. maths.anu.edu.au

[HM, 95,96]

Simple Coding Methods One of the simplest ways to encode a message is for the sender to replace every letter by another letter in an agreed manner. If the receiver of the message knows how this is done then it is usually easy to reverse the process and obtain the original message. For example, the coding method might be to replace each letter by th

X Y Z

J E B L O T S Y D V W A highly secret message such as \\the key is under the mat by the front door" would be coded as\\ofn znv hl tcqnb ofn ipo uv ofn mbaco qaab". But someone who intercepts the message might gure that \\the" is a very common word. So probably \\ofn" means \\the" and this means that \\o, f, n" are decoded as \\t, h, e" respectively. To ma

Problems with these Coding Methods

with the coding methods just discussed. There are two major problems Very soon someone will lose their copy of the code book, or sell it to some unscrupulous third party, or have it stolen. If the coding method is sent over the internet it is likely to be intercepted. If a malicious person knows how to encode a message they can reverse the process

?Working with BIG numbers

Since we will work with incredibly big numbers, let's get a feel for them. maths.anu.edu.au

Any message sent to you by RSA encoding is totally safe.48

If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au If the two members of every pair of edges point in opposite directions then the surface is orientable; If the two members of one or more pairs of edges points in the same direction then the surface is non orientable. maths.anu.edu.au

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