8th Class Mathematics Related to Competitive Exam Number System

NUMBER SYSTEM

FUNDAMENTALS

• A number r is called a rational number if it can be written in the form \[\frac{p}{q}\], where p and q are integers and \[q\ne 0.\]

Example:- \[\frac{1}{2},\frac{1}{3},\frac{2}{5}\] etc.

• Representation of Rational Number as Decimals.

• Case I :- When remainder becomes zero \[\frac{1}{2}=0.5,\frac{1}{4}=0.25,\frac{1}{8}=0.125\]

It is a terminating Decimal expansion.

• Case II :- When Remainder never becomes zero.

Example:- \[\frac{1}{3}=.3333,\frac{2}{3}=.6666,\]it is a non - terminating Decimal expansion.

• There are infinitely large rational numbers between any two given rational numbers.

• Irrational Number:- The number which cannot be expressed in form of \[\frac{p}{q}\]and neither it is terminating nor recurring, is known as irrational number.

Examples:- \[\sqrt{2},\sqrt{3}\] etc.

Rationalization :- Changing of an irrational number into rational number is called rationalization and the factor by which we multiply and divide the number is called rationalizing factor.

Example:- Rationalizing factor of \[\frac{1}{2-\sqrt{3}}\] is \[2+\sqrt{3}\].

Rationalizing factor of \[\sqrt{3}+\sqrt{2}\,is\,\sqrt{3}-\sqrt{2}\]

Low of exponents for real numbers. 

• \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
• \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
• \[{{({{a}^{m}})}^{n}}={{a}^{mn}}\]
• \[{{a}^{o}}=1\]

Some useful results on irrational number

• Negative of an irrational number is an irrational number.
• The sum of a rational and an irrational number is an irrational number.
• The product of a non - zero rational number and an irrational number is an irrational number.

Some results on square roots

• \[{{\left( \sqrt{x} \right)}^{2}}=x,x\ge 0\]
• \[\sqrt{x}\times \sqrt{y}=\sqrt{xy},\,x\ge 0\,and\,y\ge 0\]
• \[\left( \sqrt{x}+\sqrt{y} \right)\times \left( \sqrt{x}-\sqrt{y} \right)=x-y,(x\ge 0\,and\,y\ge 0)\]
• \[{{(\sqrt{x}+\sqrt{y})}^{2}}=x+y+2\sqrt{xy},(x\ge 0\,and\,y\ge 0)\]
• \[{{\left( \sqrt{x}-\sqrt{y} \right)}^{2}}=x+y-2\sqrt{xy},(x\ge 0\,and\,y\ge 0)\]
• \[\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}},(x\ge 0\,and\,y\ge 0)\]
• \[\left( a+\sqrt{b} \right)\left( a-\sqrt{b} \right)={{a}^{2}}-b,(b\ge 0)\]

\[\left( \sqrt{a}+\sqrt{b} \right)\times \left( \sqrt{c}+\sqrt{d} \right)=\sqrt{ac}+\sqrt{bc}+\sqrt{ad}+\sqrt{bd},\]\[(a\ge 0,b\ge 0,c\ge 0)\]