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REAL NUMBERS
REAL NUMBERS
The real numbers include all the rational numbers (p/q form where
q≠0), and all the irrational numbers, such as √3 and π. Real numbers
represent points on an infinitely long line called the number line or real
number line.
Here N represents Natural numbers, Z/I represent Integers, Q represents
Rational numbers and R represents Real numbers
Real numbers measure continuous variable quantities. For example
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ii.
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The temperature of Moscow at a given time
The speed of a running train.
The amount of water in a bottle.
REAL NUMBER LINE
Real number line represents a straight line with every point on it pointing
to a unique real number. Integers are marked on the line at equal
distances. Between every two integer lies infinite real numbers (fractions,
rational numbers, irrational numbers, decimals)
Points to remember
i.
There are infinite many real numbers between to real numbers or
integers
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Integers on number line are equally spaced
Real numbers include positive integers, negative integers and zero
Infinity is not a real number; it is just a notation to represent
uncountable items
All fractions and decimals (terminating or non-terminating) are
real numbers.
There is a unique real number for every number on the number
line and for every real number there is a unique point on the
number line
EUCLID’S DIVISION LEMMA
For two positive integers a and b, there exist unique integers q and r
satisfying a = bq + r, 0 ≤ r < b. e.g. 495 = 49 x 10 + 5
FUNDAMENTAL THEOREM OF ARITHMETICS
Every composite number can be factorized as a product of prime
numbers and this factorization is unique e.g. 20580 = 2 x 2 x 3 x 5
x 7 x 7 x 7. We can also say that prime factorization of a natural
number is unique.
Points to remember
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ii.
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If p is a prime and p divides x2, then p divides x, where x is a
positive integer.
Let ‘a’ be a rational number whose decimal expansion terminates,
then we can express ‘a’ in the form p/q, where p and q have no
common factor except 1(co-prime), and the prime factorisation of
q is of the form 2n5m, where n, m are non-negative integers.
Let a = p/q be a rational number, such that the prime
factorisation of q is of the form 2n5m, where n, m are non-negative
integers. Then the decimal expansion ‘a’ terminates.
Let a = p/q be a rational number, such that the prime
factorisation of q is not of the form 2n 5m, where n, m are nonnegative integers. Then ‘a’ has a decimal expansion which is nonterminating repeating (recurring)
PROPERTIES OF REAL NUMBERS
1. Commutative Property of Multiplication a • b = b • a
2. Commutative Property of Addition a + b = b + a
3. Associative Property of Addition a + (b + c) = (a + b) + c
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Associative Property of Multiplication a • (b • c) = (a • b) • c
Additive Identity Property a + 0 = a
Multiplicative identity Property a • 1 = a = 1 • a
Additive Inverse a + (-a) = 0
Multiplicative Inverse a * 1/a = 1 (a≠0)
Distributive Law a • (b + c) = ab + ac