Real number concept map

real number concept map
Real Number
Irrational number
Rational number
Terminating
Non terminating
Euclid division
Lemma
Euclid division
algorithm
Fundamental theorm
of arithmetic
X = p/q and q=0 (q is
in the form of 2 and 5 Where
m n are non negative integers ,
then it is terminating decimal)
3/22*5 = 0.15
x = p/q
and q*0
(a is not in the
form of2*5*
where m, n are
non-negative
integers, then
it is
non- terminating
decimal)
e 9
10 - 3.3
Given positive integers a
and b there exist unique
integers q and r satisfying
a = ba = r Oar
If 'a' and "b'are positive
integers such that a = bq + P,
then every common divisor
of 'a and "b' is a common
divisor of "b' and 'r', and
vice-versa.
e.g. HCF of 420 & 48
420 = 48 × 8 + 36
48 = 36 x1 + 12
36 = 12 × 3 + 0
- H.C.F, of 420 & 48 is 12
Every composite number
can be expressed or
factorized as a product of
primes
eg. 48 = 2 x 2* 2 * 2.
420 = 2 * 2 y 3 * 5*7
Application
e.g. Check whether 6* can end with the digit O for any natural
number 'n'
I Sol. Any positive integer ending with the digit zero is divisible
by 5 and so its prime factorization must contain the prime 5.
6 = (2 x 3) =2 x 3
The prime in the factorization of G' is 2 and 3
5 does not occur in the prime factorization of 6* for any n.
6 does not end with the digit zero for any natural number n
HCF
HCF. (48, 420) × L.C.M. (48,420)
= 48
> 420
(This is true for 2 numbers only)
LCM
48 = 2 x 2 x 2 x 2 x 3
420 = 2*2 x 3 x5 x7
HCF 2*2*3
eg, 48 = 2 x 2 x 2 x 2 x 3
420 = 2 x2 x3 x 5 x7
LCM = 2 y 2x 2
x 2 x 3 x 5 x 7
LCM 1680
Some Important Results
1. Let p' be a prime number and a be a positive
integer. If 'p' divides a*, then 'p' divides a
2. HCF × LCM = Prosuct of two numbers.
3. LCM is always divisible by HCF
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