maths booklet

Question 1

State algebraic properties of real numbers.

Question 2

Theorem-: (a) If z and a are elements in with z+a =a, then z=0.

(b) If u and b 0 are elements in with u.b=b,then u=1

(c) If a ,then a.0=0

Question 3

Theorem-: (a) If a 0 and b in such that a.b=1 then b=1/a.

(b) If a.b=0,then either a=o or b=0

Question 4

Theorem-: There does not exist a rational number r such that =2

Definition-: The order properties of

There is a non empty subset P of , called the set of positive real numbers

That satisfies the following properties:

(1) If a, b belong to P,then a+b belongs to P.(2) If a , b belong to P,then ab belong to P.(3) If a belong to ,then exactly one of the following is true:

a P, a=0, -a P

Definition-: let a , b be elements of .

(a) If a-b P ,then we write a>b or b<a

(b) If a-b P ,then we write a b or b a.

Prepared by sanjeev kumar shukla BSc(hons) maths and MSc maths university of delhi Contact number 9716535385

Question 5

Theorem-: let a , b , c be any elements of .

(a) If a>b and b>c , then a>c.(b) If a>b,then a+c >b+c. (c) If a >b and c>0, then ca>cb.

If a> b and c<0, then ca<cb.

Question 6

Theorem-: (a) If a and a 0, then >0.

(b)If n , then n>0.

Question 7

Theorem-: If a is such that 0 a< for every >0, then a=0.

Question 8

Theorem-: If ab>0, then either

(1) a>0 and b>0 or(2) a<0 and b<0

Question 9

(a) Let a o and b 0.Then

a < b

(b) If a and b are positive real numbers, then


(c) ,(BERNOULLI’S INEQUALITY).

Question 10

If a , b are real numbers then prove the following.

(a) If a+b=o, then a+b=0 ,then b=-a(b) –(-a)=a(c) (-1)a=-a,(d) (-1)(-1)=1(e) –(a+b)=(-a)+(-b)(f) (–a).(-b)=a.b

(g)

Question 11

If a satisfies a.a=a, prove that either a=o or a=1.

Question 12

,show that.

Question 13

Show that there does not exist a rational number s such that =6

Question 14

Show that there is no rational whose square is 3.

Question 15

Show that sum and product of two rational numbers, is a rational number.


Question 16

(a) Prove that if x is rational and y is irrational then x+y is irrational.(b) If x is non zero rational and y is irrational then x.y is irrational.(c) Show that there is no natural number between 0 and 1(d) Prove that no natural number can be both even and odd

(e) If c>1 and m , n are natural numbers ,show that .

(f) If 0<c<1 and m , n are natural numbers,show that

Question 17

(a) If a<b and c d, prove that a+c<b+d.(b) If 0<a<b and 0 c d,prove that 0 a.c b.d.

(c) Show that if a>0 , then >0 and .

(d) Show that if a<b , then .

Question 18

(a) If 0<c<1,show that 0<

(b) If 1<c, show that .

(c) If

Absolute value and real line

Definition-:The absolute value of a real number a, denoted by ,is defined by


Question 19

Theorem- : (a)

Question 20

Theorem-:(Triangle inequality)

If .

Question 21

Corollary-: If


Question 22

Definition-:

Question 23

Theorem-:

.

Question 24

Question 25


Question 26

Qestion 27

.

Question 28

.

Question 29

Question 30

Question 31


Question 32

Question 33


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