Revisiting rational numbers and their decimal expansion

CBSE Class 10 Mathematics-Real Numbers- Revisiting rational numbers and their decimal expansion Notes with Examples. Get detailed CBSE study material for class 10 maths.

Revisiting rational numbers and their decimal expansion

 

 

 

Revisiting Rational Numbers

Rational Numbers and their Decimal Expansion

The decimal expansion of every rational number is either terminating or non-terminating repeating.

Terminating Decimal Expansion

The number which terminates after a finite number of steps in the process of division is called terminating decimal expansion. E.g. 6.25, 1.14

Non-Terminating Decimal Expansion

The number which does not terminate in the process of division is called non-terminating decimal expansion.

There are two types of non-terminating decimal expansions

a) Non-terminating repeating Decimal Expansion

The number which does not terminate but repeats the particular number again and again in the process of division is said to be non-terminating repeating decimal.

E.g. 0.333333

b) Non-terminating Non-repeating Decimal Expansion

The number which neither terminates nor repeats the particular number in the process of division is said to be non-terminating repeating decimal. E.g. 1.03303033

Theorem 1:

Let x be a rational number whose decimal expansion terminates.

Then x can be expressed in the form {\color{Red} \frac{p}{q}} , where p and q are coprime and the prime factorization of q is of the form 2m5, where m, n are non- negative integers.

Theorem 2:
Let x = {\color{Red} \frac{p}{q}} be a rational number, such that the prime factorization of q is of the form 2m5n , where m, n are non-negative integers, then {\color{Red} \frac{p}{q}} has a terminating decimal.

{\color{Blue} \frac{23}{80}=\frac{23}{2^{4}X5}}

Multiplying both numerator and denominator by {\color{Blue} 5^{3}}

{\color{Blue} \frac{23 X 5^{3}}{2^{4}X5X5^{3}}} = {\color{Blue} \frac{23 X 5^{3}}{2^{4}X5^{4}}}

{\color{Blue} \frac{2875}{(2X5)^{4}}} = {\color{Blue} \frac{2875}{(10)^{4}}} = {\color{Blue} \frac{2875}{10000}} = 0.2875

Theorem 3:
Let x ={\color{Red} \frac{p}{q}}be a rational number, such that the prime factorization of q is not of the form 2m5, where m, n are non-negative integers, then x has decimal expansion which is non-terminating repeating.

Example:
Without actually performing the long division, state whether {\color{Red} \frac{543}{225}}  has a terminating decimal expansion or non-terminating recurring decimal expansion.

                                                               {\color{Blue} \frac{543}{225}= \frac{181}{75}=\frac{181}{3 X 5^{2}}}

Since the factors of the denominator {\color{Blue} (3.5^{2})} is not of the form 2m5n

So,{\color{Blue} \frac{543}{225}} is non-terminating recurring decimal expansion.

 

Example:
The decimal expansion of the rational number{\color{Red} \frac{53}{2^{4}X 5^{3} }} will terminate after how many places of decimal.

                                                             {\color{Blue} \frac{53}{2^{4} X 5^{3}}}

Now, multiplying both numerator and denominator by 5

                        {\color{Blue} \frac{23 X 5}{2^{4}X5^{3}X5}=\frac{265}{2^{4}X5^{4}}}

              = {\color{Blue} \frac{265}{(2 X 5)^{4}}= \frac{265}{10^{4}}\frac{265}{10000}=0.0265}

The decimal expansion of the rational number {\color{Blue} \frac{53}{2^{4}X5^{3}}} will terminate after 4 places of decimal.

Example:If {\color{Red} \frac{299}{4000}} = {\color{Red} \frac{299}{2^{m}5^{n}}}, then find the value of m and n, where m and n are non- negative integers. Hence, write its decimal expansion without actual division.

{\color{Blue} \frac{299}{4000}=\frac{299}{2^{m}5^{n}}}

{\color{Blue} \frac{299}{4000}=\frac{299}{2^{5}5^{3}}}

{\color{Blue} \frac{299}{2^{m}5^{n}}} = {\color{Blue} \frac{299}{2^{5}X5^{3}}}

m = 5 and n = 3

Now,{\color{Blue} \frac{299}{4000}} can be expressed as {\color{Blue} \frac{299}{2^{5}X5^{3}}}

We have{\color{Blue} \frac{299}{2^{5}X5^{3}}}

Multiplying both numerator and denominator by {\color{Blue} 5^{2}}

{\color{Blue} \frac{299 X 5^{2}}{2^{5}X5^{3}X5^{2}}= \frac{7475}{2^{5}X5^{5}}}

{\color{Blue} \frac{7475}{(2X5)^{5}}= \frac{7475}{10^{5}}= \frac{7475}{100000}= 0.07475}

 

Example:

What can you say about the prime factorization of the denominators of

the following rational:

i) 42.123456789

ii) {\color{Red} \overline{32.56789}}

i) Since 42.123456789 has a terminating decimal expansion. So, its denominator is of the form 2m5, where m and n are non-  negative integers.

ii) Since {\color{Blue} \overline{32.5678}}  has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

Revisiting rational numbers and their decimal expansion

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