Hg
Name: Class 10 STICK TO YOUR WALL IN STUDY AREA
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The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
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REAL NUMBERS
• Real Numbers are classified as follows:
Real Numbers
Rational (are in p/q form) Irrational (not in p/q form)
(Decimal form is either terminating or recurring) (Decimal form is neither terminating nor recurring)
(e.g. 2, 3, 7, )
Terminating Recurring
(q is of the form 2
n
5
m
) (q is not of the form 2
n
5
m
)
Integers, Whole, Natural (e.g. …-2, -1, 0, 1, 2…) Fractions (e.g. 22/7, 1/3, 5/33, 7/30)
& Fractions (e.g 1/2, 33/10, 7/20)
• Euclid’s division lemma: Given positive integers a and b, there exist whole number q and r such that a
= bq + r and 0 ≤ r < b.
• Euclid’s division algorithm: This is based on Euclid’s division lemma is used to find the HCF of any two
positive integers a and b, with a > b as follows:
o Divide a by b to find q and r (quotient and remainder)
o If r = 0, then the HCF is b, otherwise, apply Euclid’s lemma to b and r.
o Continue this process till remainder is zero.
o The last divisor is the HCF of (a, b) and is also the HCF of intermediate dividend and divisor pairs.
• Fundamental theorem of Arithmetic: Every composite number can be expressed (factorised) as a
product of prime numbers, and factorisation is unique apart from the order in which the prime factors
occur.
o LCM (a, b) = product of highest powers of all prime factors of a and b.
o HCF (a, b) = product of lowest powers of all prime factors of a and b.
o HCF of a, b * LCM of a, b = a * b
• If p is a prime number and p divides a
2
, then p divides a, where a is a positive integer. Using this
theorem, we prove 2, 3 etc. to be irrational numbers.
• Let x be a rational number expressed in p/q form where p and q are co-prime, its decimal form will be:
o terminating if prime factorisation of q is of the form 2
n
5
m
o recurring if prime factorization of q is NOT of the form 2
n
5
m
, where n and m are whole
numbers.
• is irrational whereas 22/7 is rational. Since = 3.141592…. and 22/7 is 3.142857142857.…, 22/7 is
just a good approximation of we use in calculations.
