chapter 1 class 10 number system
class 10 maths
Chapter -1 (REAL NUMBERS)
Natural Numbers:
counting numbers are known as natural numbers.
Eg. 1,2,3,4…….
Whole numbers:
counting
numbers including zero are known as whole numbers.
i.e. 0, 1, 2, 3, 4, 5, …………….
Integers:
All whole numbers and natural numbers numbers altogether known as integers.
i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
Composite Numbers:
The numbers which has more
than two factors are called composite numbers.
NOTE : “1” is neither a prime number nor a
composite number. It is a unit.
RATIONAL NUMBERS: The number which can be put in the form of p|q are called rational
numbers . eg 7/5 , 5/4, ½ …etc.
IRRATIONAL NUMBERS :
The number which cannot be
put in the form of p/q are called irrational numbers.
REAL NUMBERS:
All the rational &
irrational numbers are called real number.
PRIME NUMBERS:
The number which has only
two factors one 7 itsrlf are called prime numbers. Eg. 2, 3, 5, 7, 11, 13….etc.
CO-PRIMES
If the H.C.F of two numbers
is 1 then number are co –prime numbers. Eg
(5,7)
(13, 27) ,(15,16)….etc
EVEN NUMBERS
Natural numbers which are divisible by two are
called even numbers. Eg 2,4,6,8,10…….
ODD NUMBERS
Natural numbers which are
not divisible by 2 are called odd numbers. Eg 1,3,5,7,…….
TWIN PRIMES
Consecutive prime numbers
which are differ by 2 are called twin primes . eg (5,7), (11,13),
(17,21)…….etc.
IMAGINARY NUMBER OR Non – REAL NUMBERS
Negative suare root of a
natural numbers is called imaginary numbers eg. √5,√3,√7….etc.
Euclid’s Division
Lemma
For
each pair of given positive integers a and b, there exist unique whole numbers q and r which
satisfies the relation
a
= bq + r, 0 ≤ r < b, where q and r
can also be Zero.
where ‘a’ is a dividend, ‘b' is divisor, ‘q’ is
quotient and ‘r’ is remainder.
∴ Dividend = (Divisor x Quotient) + Remainder
Algorithm
An algorithm gives us some definite steps to solve a
particular type of problem in a well-defined manner.
Lemma: A lemma is a
statement which is already proved and is used for proving other statements.
Euclid’s Division
Algorithm
This concept is based on Euclid’s division lemma. This
is the technique to calculate the HCF (Highest common factor) of given two
positive integers m and n,
To calculate the HCF of two positive integers’ a and b
with a > b, the following steps are followed:
Step 1: Apply Euclid’s
division lemma to find q and r where a =
bq + r, 0 ≤ r < b.
Step 2: If the
remainder i.e. r = 0, then the HCF will be ‘b’ but if r ≠ 0 then we have to apply
Euclid’s division lemma to b and r.
Step 3: Continue with
this process until we get the remainder as zero. Now the divisor at this stage
will be HCF(a, b). Also, HCF (a, b) = HCF (b, r), where HCF (a, b) means HCF of
a and b.
The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product
of primes & this factorization is
unique
·
HCF product of
common factors with smallest power.
·
LCM product of
all factors with greatest power.
HCF (m, n) × LCM (m, n) = m × n
i.e HCF ×LCM= PRODUCT OF TWO
NUMBERS
Rational Numbers
The number ‘s’ is known as a rational number if
we can write it in the form of m/n
where ‘m' and ‘n’ are integers and n ≠ 0, 2/3, 3/5 etc.
Rational numbers can be written in decimal form also
which could be either terminating or non-terminating. E.g. 5/2 = 2.5 (terminating) and
(non-terminating).
Irrational Numbers
The number ‘s’ is called irrational if it cannot be
written in the form of m/n, where m and n are integers and n≠0 or in the
simplest form, the numbers which are not rational are called irrational
numbers. Example - √2, √3 etc.
·
If p is a prime
number and p divides a2 , then p is
one of the prime factors of a2 which
divides a, where a is a positive integer.
·
If p is a
positive number and not a perfect square, then √n is
definitely an irrational number.
·
If p is a prime
number, then √p is also an irrational
number.
Rational Number and their Decimal Expansions
·
Let y be a real
number whose decimal expansion terminates into a rational number which we can
express in the form of a/b, where a and b are coprime, and the prime
factorization of the denominator b has the powers of 2 or 5 or both like 2n5m, where n, m are
non-negative integers.
·
Let y be a
rational number in the form of y = a/b, so that the prime factorization of the
denominator b is of the form 2n5m, where n, m
are non-negative integers then y has a terminating decimal expansion.
·
Let y = a/b be a
rational number, if the prime factorization of the denominator b is not in the
form of 2n2m, where n, m are
non-negative integers then y has a non-terminating repeating decimal expansion.
o
The decimal
expansion of every rational number is either terminating or a non-terminating
repeating.
o
The decimal form
of irrational numbers is non-terminating and non-repeating.
