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Gyaan Maths
CBSE · Class 6-12
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Class 9 · Ganita Manjari
Unit 3 · Class 9Ganita Manjari
The World of Numbers
Rationals, irrationals, real numbers, and decimal expansions.
Lessons
4 concepts
1
Rationals vs Irrationals
p
/
q
p/q
p
/
q
in lowest form: terminates iff denominator has only 2s and 5s as prime factors. Irrationals include
2
,
3
,
π
,
e
\sqrt 2, \sqrt 3, \pi, e
2
,
3
,
π
,
e
.
2
Density of Reals
Between any two reals lie infinitely many rationals AND infinitely many irrationals.
3
Rationalising the Denominator
1
a
+
b
⋅
a
−
b
a
−
b
=
a
−
b
a
2
−
b
\dfrac{1}{a+\sqrt b}\cdot \dfrac{a-\sqrt b}{a-\sqrt b} = \dfrac{a-\sqrt b}{a^2-b}
a
+
b
1
⋅
a
−
b
a
−
b
=
a
2
−
b
a
−
b
. Removes the surd.
4
Exponent Laws
a
m
⋅
a
n
=
a
m
+
n
a^m\cdot a^n = a^{m+n}
a
m
⋅
a
n
=
a
m
+
n
;
a
−
n
=
1
/
a
n
a^{-n} = 1/a^n
a
−
n
=
1/
a
n
;
a
1
/
n
=
a
n
a^{1/n} = \sqrt[n]{a}
a
1/
n
=
n
a
.
Skills to practise
Skill · medium
The World of Numbers — Practice