Ammoi Maths

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FACULTY OF EDUCATION AND LANGUAGE STUDIES

SEMESTER MAY 2011

SBMA 4403

ELEMENTARY ANALYSIS/PENGANTAR ANALISIS

STUDENT NAME : PUVANESVARY A/P JEGANATHAN

MATRICULATION NUMBER : 790709055666001

IDENTITY CARD NUMBER : 790709-05-5666

TELEPHONE NUMBER : 0192077457

E-MAIL : [email protected]

LEARNING CENTRE : SHAH ALAM


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TABLE OF CONTENT

Question 1 a ...................................................................................... 2

Question 1 b .................................................................................... 2

Question 1 c .................................................................................... 3

Question 1 d .................................................................................... 4

Question 2 .................................................................................... 6

References ................................................................................... 7

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1. Prove the following statements.

a) 0 is an even number.

A number is called even if it is an integer multiple of 2. Zero is an integer

multiple of 2, namely 0 2, so zero is even

b) Ifa is odd then a2+ a is even.

Let a is odd. Then, a2 is also will odd.

For example, 12 = 1

32 = 9

52 = 25

When a odd number added with odd number, we obtain even number

For example, 3 + 3 = 6

5 + 3 = 8

Therefore, it is proved that, if a is odd then a2

+ a is even.

c) Square root of 3 is irrational.

To prove that this statement is true, let us assume that is rational so that we may write

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.................................................(1)

for a and b = any two integers. We must then show that no two such integers can be found.

We begin by squaring both sides of eq.(1):

b

a=3

22

2

2

3or3 abb

a

==

:haveweand2p,bathato

even.bathen,odd

odd

b

a3f

.odd

oddor

even

even

b

a3abovediscussedisitAs

22

222

2

2

2

=+

=+==

==

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(2m + 1)2 + (2n + 1)2 = 2p

4m2 + 4m + 1 + 4n2 + 4n + 1 = 2p

4m2 + 4m + 4n2 + 4n + 2 = 2p

2m2 + 2m + 2n2 + 2n + 1 = p where p is odd

ion!contradictaeven,bemustpso,b

p2

b

2p4

bba4

1b

a1

b

a3

2

2

2

22

2

2

2

2

=

=

+

=

+=+

=

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Thus, the square root of 3 cannot be a rational number, so it is an irrational.

d) If n is even and 4 n 30, then n can be written as a sum of two prime numbers.

According Goldbach Conjeture, it is stated clearly that

"Every even integer greater than 2 can be written as the sum of two primes"By using the statement we can prove it inductively such as below

6 = 3 + 3

8 = 5 + 3

10 = 7+ 3

.......

30 = 23 + 7

Therefore, it is proved that, ifn is even and 4 n 30, then n can be written as a

Sum of two prime numbers.

e) There exist infinitely many rational numbers between

two rational numbers.

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Question 2

Prove that . Hence, prove that .

We can express 2 in the following form:

where a and b are integers with no common factors, as required by our definition of a

rational number, as discussed above.

numbers.rationalany twobetweennumberational

oneleastatalwaysisthereTherefore,.irrationalalsoisnumberrationalalus

numberirrationalanagain,Butz.aisorigin,thex tofromdistancehe

.irrationalisz,iswhichx,toafromdistanceThereforenumber.irrationaln

alsoisnumberrationalax)case(in thisnumberirrationalanut

)q

p-(x

q

p-xa-

:So,aandbetween xdistancethebezet

.q

pformin thenumbers,rationalbotharebnandanhere

],b,[aIintervalin thenumberirrationalanbeet x

n

n

1

1

1

1n

n

nnn

+

+

+==

=

Q2 Q++ 11 nn Nn

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Now, the square of an even number is even, and the square of an odd number is odd.

The square of the even number 2n is 4n2, which is clearly even, and the square of the

odd number 2n+1 is 4n2+4n+1, which is clearly odd. With this in mind, we note that a

must be even and hence we can write

where c is another integer. Now, substituting this into the previous expression gives

This shows that b is also an

even number, and hence

where dis yet another integer.Hence, it is prove that

By using the same idea, we look into

Q2

Q++ 11 nn

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( )

fact.followingtheprovecanwef

completebewillproofThisrational.is1-nthathavewe

Q,oflawsclosureby therational,arenand2sinceNext,

rational.alsois1-n*22n1n1nhen

rational.is1n1nthatuppose

1.nthatuppose

.irrationalshowneasilyiswhich,21n1nthen1,nf

2

22

+=++

++

>

=++=

9

NnforQ1n1nthatprovedalsoisitHence,

squares.perfect

econsecutivobetween twis1-nthatshowsrootssquaretaking

1,nintegersallforn1-n1)-(nsinceprecisely,More

1.2nn-1)(nThen

1.naslongasapartunit1thanmorespaced

aresquareseconsecutivthatclaimthefromfollowsThis

.irrationalis1-n

2

222

22

2

2

++

>


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Reference

Bourbaki, N. lments de mathmatique: Algbre. Reprinted as Elements of Mathematics:

Algebra I, Chapters 1-3. Berlin: Springer-Verlag, 1998.

Courant, R. and Robbins, H. "The Rational Numbers." 2.1 in What Is Mathematics?: An

Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University

Press, pp. 52-58, 1996.

Finch, S. R. "Powers of 3/2 Modulo One." 2.30.1 in Mathematical Constants. Cambridge,

England: Cambridge University Press, pp. 194-199, 2003.

Honsberger, R.More Mathematical Morsels.Washington, DC: Math. Assoc. Amer., pp. 52-53,

1991.

Salamin, E. and Gosper, R. W. Item 54 in Beeler, M.; Gosper, R. W.; and Schroeppel, R.

HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 18,

Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item54.

Wolfram, S.A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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