32355118
Description
Mind Map by Aryan Bhatt, updated more than 1 year ago
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Created by Aryan Bhatt
almost 3 years ago
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Real numbers
- Theorems
- Statement
- 1. Let p be a prime no. If p divides a²
. Then p divides a,where a is a
positive integer.
- 2. √2 is irrational
- 3. Let x be a rational no. Whose
decimal expansion terminates .
Then x can be expressed in the
form of p/q, where p &q are co
prime, the prime factorisation of q
is of the form 2ⁿ, 5 raise to power m
, where n,m are non- negative
integers.
- 4. Let x = p/q be a rational number
such that the prime factorisation of q
is of the form 2ⁿ, 5 raise to power m .
Where n, m are non - negative
integers. Then x has a decimal
expansion which terminates.
- 5. Let x =p/q be a rational number, such
that the prime factorisation of q is not
of the form of 2ⁿ5 raise to power m .
Where n, m are non - negative integers.
Then x has a decimal expansion which
is non - terminating repeating.
- 5. Let x =p/q be a rational number, such
that the prime factorisation of q is not
of the form of 2ⁿ5 raise to power m .
Where n, m are non - negative integers.
Then x has a decimal expansion which
is non - terminating repeating.
- 4. Let x = p/q be a rational number
such that the prime factorisation of q
is of the form 2ⁿ, 5 raise to power m .
Where n, m are non - negative
integers. Then x has a decimal
expansion which terminates.
- 3. Let x be a rational no. Whose
decimal expansion terminates .
Then x can be expressed in the
form of p/q, where p &q are co
prime, the prime factorisation of q
is of the form 2ⁿ, 5 raise to power m
, where n,m are non- negative
integers.
- 2. √2 is irrational
- 1. Let p be a prime no. If p divides a²
. Then p divides a,where a is a
positive integer.
- Statement
- Euclid
- Given positive integers are a, b. There
exist unique integers q and r.
Satisfying a= bq +r ; 0 5 r < b
- Division algorithm.
- Steps to obtain the HCF of
two positive integers. Say
c and d, with c > d.
- Step 1- apply euclid
division lemma, to c and d,
c=dp +r.
- Step 2- if r = zero, d is the
HCF of c and d. If r is not
equal to zero, apply euclid
division to d and r.
- Step 3- continue the
process till the
remainder is zero.
- Step 3- continue the
process till the
remainder is zero.
- Step 2- if r = zero, d is the
HCF of c and d. If r is not
equal to zero, apply euclid
division to d and r.
- Step 1- apply euclid
division lemma, to c and d,
c=dp +r.
- Steps to obtain the HCF of
two positive integers. Say
c and d, with c > d.
- Division lemma.
- Given positive integers are a, b. There
exist unique integers q and r.
Satisfying a= bq +r ; 0 5 r < b
- Fundamental theorem of airthematic
- Every composite no. Can be expressed as a
product of primes, and this factorisation is
unique, apart from the order in which the
prime factors occur
- Composite no. X=p₁*p₂*p₃..*p₄ where
p₁p₂..px are prime no.
- Composite no. X=p₁*p₂*p₃..*p₄ where
p₁p₂..px are prime no.
- Every composite no. Can be expressed as a
product of primes, and this factorisation is
unique, apart from the order in which the
prime factors occur
- prime factorisation method
- for any two positive inteɡers , a and b
- HCF (a,b)*LCM(a,b)=a*b
- for example
- f(x)=3x²y
- ɡ(x)=6xy²
- HCF=3xy
- LCM=6x²y²
- LCM=6x²y²
- HCF=3xy
- ɡ(x)=6xy²
- f(x)=3x²y
- for example
- HCF (a,b)*LCM(a,b)=a*b
- for any two positive inteɡers , a and b
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