Numbers and Proofs
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Numbers and Proofs

Reg Allenby

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  2. English
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eBook - PDF

Numbers and Proofs

Reg Allenby

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About This Book

'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow.Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths.

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Information

Publisher
Butterworth-Heinemann
Year
1997
ISBN
9780080928777
Topic
Mathématiques
Subtopic
Mathématiques générales
10
NUMBERS 
AND 
PROOFS 
Figure 
1.1 
All 
proofs 
must 
start 
somewhere 
Figure 
1.1 
is 
actually 
very 
useful 
for 
making 
two 
points 
quite 
forcefully. 
First 
we 
ask: 
do 
the 
assertions 
denoted 
by 
A, 
B, 
C, 
H 
and 
MI 
have 
some 
kind 
of 
universal 
quality? 
Don't 
they 
depend 
on 
anything 
at 
all? 
Can 
they 
not 
be 
deduced 
from 
anything 
more 
'basis'? 
In 
fact, 
in 
Figure 
1.1, 
B 
represents 
the 
assertion 
that 
l
p
= 
1, 
so 
one 
could 
argue 
that 
B 
is 
a 
consequence 
of 
the 
equality 
1-1 
= 
1 
(and 
MI). 
Then 
you 
may 
ask: 
what 
is
1· 
1
= 
1 
a 
consequence 
of? 
What, 
indeed, 
is 
meant 
by 
T? 
(As 
we 
are 
now 
getting 
a 
bit 
too 
philosophical 
we 
take 
this 
no 
further.) 
It 
is 
clear 
that 
we 
cannot 
keep 
regressing 
in 
this 
way 
for 
ever. 
Accordingly, 
in 
the 
proofs 
that 
we 
look 
at 
later, 
we 
shall 
be 
happy 
to 
accept 
certain 
easily 
believed, 
simple 
assertions 
as 
being 
unquestionably 
true 
(for 
example, 
that 
l
20
= 
1) 
and 
not 

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