Real numbers:
Real numbers are the numbers which include both rational and irrational numbers. A real number is denoted by \(\mathbb{R}\).
Important!
Every real number is either rational or irrational.
German mathematicians, Cantor and Dedekind, discovered that 'Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number'.
Decimal expansion of rational numbers:
Rational numbers can be expanded in the form of decimals by doing the usual long division.
A terminating decimal is a decimal number that has the finite number of digits the decimal point.
A recurring decimal is a decimal number that has repeating number/numbers which continues infinitely.
Now we will analyse the decimal expansions of different types of rational numbers.
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Rational number
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Value with the help of long division
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Nature of the decimal expansion
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Terminating decimals | |
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Recurring and non-terminating decimals |
Thus, it can be concluded as follows:
A rational number can have either a terminating decimal expansion or a non-terminating recurring decimal expansion.
Predicting the type of decimal expansion:
The decimal expansion of a rational number \(\frac{p}{q}\), where \(q \neq 0\) in its lowest form will be terminating when the prime factors of \(q\) are only \(2\) or only \(5\) or both \(2\) and \(5\).
Example:
The decimal expansion of \(\frac{2}{5}\) will terminate as the denominator has the prime factor \(5\).
The value of \(\frac{2}{5} = 0.4\)
Converting a decimal number to fractional form:
Case - (I): The decimal expansion is purely repeating (Eg: \(1.\overline{34}\)).
Step - 1: Consider the given decimal number as \(x\).
Step - 2: Multiply \(x\) by \(10^{n}\) where \(n\) is the number of repeating digits.
Step - 3: Subtract the original number from the number obtained from step 2, then solve for \(x\).
Case - (II): The decimal expansion is general repeating (Eg: \(1.15\overline{34}\)).
Step - 1: Consider the given decimal number as \(x\).
Step - 2: Multiply \(x\) by \(10^{m}\) where \(m\) is the number of non-repeating digits.
Step - 3: Multiply \(x\) by \(10^{n}\) where \(n\) is the number of repeating digits.
Step - 4: Subtract the numbers obtained from previous two steps and solve for \(x\).
Decimal expansion of irrational numbers:
Let us analyse the numbers with non terminating and non- recurring decimal expansion.
Example:
Let us find the square root of .
Thus, the decimal expansion of have non-terminating and non-recurring decimals.
It goes like
The decimal expansion of an irrational number is non-terminating and non-recurring. Conversely, the decimal expansion of a number is non-terminating and non-recurring is an irrational number.