Ch 1 extends the number system from rationals to real numbers by introducing irrational numbers. Students learn about decimal expansions, representing irrational numbers on the number line, operations on real numbers, and laws of exponents.
Rational numbers have terminating or repeating decimal expansions (e.g., 1/4 = 0.25, 1/3 = 0.333…). Irrational numbers have non-terminating, non-repeating decimal expansions (e.g., √2, π). Together, rationals and irrationals form the real numbers.
Every real number corresponds to a unique point on the number line. Irrational numbers like √2 can be located using the Pythagoras theorem: construct a right triangle with legs 1 and 1, then the hypotenuse is √2.
To rationalise 1/√a, multiply numerator and denominator by √a. To rationalise 1/(√a + √b), multiply by the conjugate (√a − √b). This removes surds from the denominator.
For positive real numbers a, b and rationals p, q: aᵖ × aᵍ = aᵖ⁺ᵍ, (aᵖ)ᵍ = aᵖᵍ, aᵖ/aᵍ = aᵖ⁻ᵍ, aᵖ × bᵖ = (ab)ᵖ.
Download: https://ncert.nic.in/textbook/pdf/iemh101.pdf | Complete book: https://ncert.nic.in/textbook/pdf/iemh1ps.zip
π is irrational. Its decimal expansion is non-terminating and non-repeating: 3.14159265358… The common approximation 22/7 is only an approximation, not equal to π. π cannot be expressed as p/q for any integers p, q.
Rational numbers can be written as p/q (q ≠ 0) and have terminating or repeating decimals. Irrational numbers cannot be expressed as p/q — their decimal expansions are non-terminating and non-repeating.
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