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Real Numbers

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by

Anshul Gupta

on 27 September 2014

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Transcript of Real Numbers

Natural Numbers
1, 2, 3, 4, 5...............
All numbers which are...well... "Natural"!!!
Real Numbers
Whole Numbers
0
+
1, 2, 3, 4, 5....
Natural Numbers
Just add the "hole"!!!
0, 1, 2, 3, 4, 5......
Whole Numbers
......-5, -4, -3, -2, -1
+
Integers
Rational Numbers
Any number which can be written in the form of
p/q {p,q = integers; q = 0}

Ex. 2/3; 4; 0; 7/2; 6547/2143......
Irrational Numbers
A number whose exact value is a:

Non-Terminating & Non-Repeating Decimal
Ex. 0.001000100001.... ; √2, 3, π
Types of Irrational Numbers
1. Non-terminating, non-repeating
Ex. 0.313113111311111...; 5.252252225...; 0.554445554444
2. √m; where m is not a perfect square
Ex. √2, √3, √5,√53......
3. m; where m is not a perfect cube
Ex. 2, 3, 5, 53......
4. The Number π (pie). Its approx. value is 22/7. π has a value which is non-terminating, non-repeating.
/
Properties of Real Numbers
Addition Properties
Multiplication Properties
Addition Properties
1. Closure Property
Real Number 1 + Real Number 2 = Real Number 3
2. Associative Law
(a + b) + c = a + (b + c)
3. Commutative Law
a + b = b + a
4. Existence of Additive Identity
a + 0 = 0 + a = a
5. Existence of Additive Inverse
a + (-a) = (-a) + a = 0
1. Closure Property
Real Number 1 x Real Number 2 = Real Number 3
2. Associative Law
(ab)c = a(bc)
3. Commutative Law
ab = ba
4. Existence of Multiplicative Identity
a . 1 = 1 . a = a
5. Existence of Multiplicative Inverse
a . (1/a) = (1/a) . a = 1
Multiplication Properties
Rationalisation
Rationalising Factor (RF)
If product of two irrational number is rational, then they are called the rationalising factor of each other
Ex. (a + b) and (a - b) are RF of each other.
Q. Find a rational number between 1/4 and 1/3.
Exercise
Q. Find five rational number between 3/5 and 4/5.
Exercise
Exercise
1. Add (3√2 + 7 √3) and (√2 - 5√3)
3. Simplify: (√13 - √6)(√13 + √6)
2. Divide 15√15 by 3√5
Exercise
1. Rationalise the denominator of:
a. 1 / (3 + √5)
b. 4 / (√7 + √2)
c. (√4 + √3) / (√4 - √3)
Exercise
2. If a and b are rational numbers and

(4 + 3√5) / (4 - 3√5) = a + b√5,

Find the value of a and b.
Class IX Refresher
Numbers whose square is non-negative
Laws of Radicals
For a real no. a > 0 and rational numbers
p
&
q
i) a x a = a
p
q
(p + q)
ii) (a ) = a
p
q
pq
iii) a / a = a
p
q
(p - q)
iv) a x b = (ab)
p
p
p
Exercise
Full transcript