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Maths: Real Numbers
Points to Remember
• When the decimal expansion of p/q , q≠0 terminates that is, comes to an end, the decimal is called a terminating decimal.
• In the decimal expansion of p/q , q≠0 when the remainder is not zero, we have a repeating (recurring) block of digits in the quotient. In this case, the decimal expansion is called non-terminating and recurring.
• If a rational number p/q , q≠0 can be expressed in the form , where p ∈ Z and m, n ∈ W, then the rational number will have a terminating decimals. Otherwise, the rational number will have a non-terminating repeating (recurring) decimal.
• A rational number can be expressed either a terminating or a non- terminating recurring decimal.
• An irrational number is a non-terminating and non-recurring decimal, i.e. it cannot be written in form p/q , where p and q are both integers and q ≠ 0.
• The union of all rational numbers and all irrational numbers is called the set of real numbers.
• Every real number is either a rational number or an irratonal number.
• If a real number is not rational number, then it must be an irrational number.
• If ‘a’ is a positive rational number, ‘n’ is a positive integer and if n√a is an irrational number, then n√a is called as a surd.
• If ‘m’, ‘n’ are positive integers and a, b are positive rational numbers, then
(i) ( n√a)n = a = n√an (ii) n√a × n√b = n√ab (iii) m√n√a = mn√a = n√m√a (iv) n√a / n√a = n√[a/b]
• The process of multiplying a surd by another surd to get a rational number is called Rationalisation.
• Expressing a number N in the form of N = a ×10n where, 1 ≤ a < 10 and ‘n’ is an integer is called as Scientific Notation.
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