31250944
Description
Mind Map by JUDHAJEET GHOSH, updated more than 1 year ago
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Created by JUDHAJEET GHOSH
over 3 years ago
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Real Numbers
- Euclid's Division Lemma
- any Positive integer 'a' can be
divided by positive integer 'b' such
that remainder 'r'< 'b' is found.
[theorem 1.1]
- Simply Long
Division
Method
- Algebraically: a=bq+r, 0 ≤r≥ b
- Algebraically: a=bq+r, 0 ≤r≥ b
- Simply Long
Division
Method
- Application:
computing HCF of 2
positive integers
- to obtain HCF: c,d are the
numbers,c+dq+r,0 ≤r≥ b,if 'r'=0 then 'd' is
the HCf else continue the process with 'd'
and 'r' until remainder=0,devisor in that
stage is the HCF
- to obtain HCF: c,d are the
numbers,c+dq+r,0 ≤r≥ b,if 'r'=0 then 'd' is
the HCf else continue the process with 'd'
and 'r' until remainder=0,devisor in that
stage is the HCF
- any Positive integer 'a' can be
divided by positive integer 'b' such
that remainder 'r'< 'b' is found.
[theorem 1.1]
- Fundamental Theorem of Arithmetic
- Every Composite Number can
be Expressed as a Product of
Primes in a Unique way. [theorem 1.2]
- Application
- Proving
Irrationality of
Numbers
- When 'p' is a Prime no.
and 'p' divides 'a²' then 'p'
also divides 'a'. [theorem 1.3]
- When 'p' is a Prime no.
and 'p' divides 'a²' then 'p'
also divides 'a'. [theorem 1.3]
- Finding
Exact
Decimal
Expansion
of
a
Rational
Number
- if 'x' is Rational No. Whose decimal Expansion is
Terminating,Then it can be Expressed as 'p/q',
where "p" and 'q' are co primes and factors of 'q' is
in form 2^n*5^m,and 'n'&'m' are non-negative
integers. [theorem 1.5]
- If prime factorizing 'q' gives non 2^n*5^m form
then the Decimal Expansion is non-terminating and
repeating. [theorem 1.7]
- If prime factorizing 'q' gives non 2^n*5^m form
then the Decimal Expansion is non-terminating and
repeating. [theorem 1.7]
- if 'x' is Rational No. Whose decimal Expansion is
Terminating,Then it can be Expressed as 'p/q',
where "p" and 'q' are co primes and factors of 'q' is
in form 2^n*5^m,and 'n'&'m' are non-negative
integers. [theorem 1.5]
- HCF and LCM of
two numbers by
Prime
Factorization
- Proving
Irrationality of
Numbers
- Here 'expressed'
means factorized
and 'unique' doesn't
include the order of
factors
- Application
- Every Composite Number can
be Expressed as a Product of
Primes in a Unique way. [theorem 1.2]
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