Category :
9th Class
NUMBER SYSTEM
FUNDAMENTAL
- A number which can be expressed in the form of \[\frac{p}{q}\], Where p and q are integers and \[q\ne 0\]is called a rational number.
Example:- \[\frac{1}{2},\frac{1}{3},\frac{2}{5}\] etc.
Representation of Rational Number as Decimals.
- Case I:- When remainder becomes zero \[\frac{1}{2}=.5,\frac{1}{4}=.25,\frac{1}{8}=.125\] 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 rational numbers between any two given rational numbers.
- Irrational Number: The number which cannot be part in form of \[\frac{p}{q}\]and neither there are terminating nor recurring are known as irrational Number.
Example:- \[\sqrt{2},\sqrt{3}\text{ }etc.\]
- 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\[\sqrt{3}-\sqrt{2}\]
LAW OF EXPONENTS FOR REAL NUMBERS
- \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
- \[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}\]
- \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]
- \[{{a}^{{}^\circ }}=1\]
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
Some results on square roots
- \[{{\left( \sqrt{x} \right)}^{2}}=x,x\ge 0\]
- \[\sqrt{x}\times \sqrt{y}=\sqrt{xy},\,\,x\ge 0\,and\,y\ge 0\]
- \[\left( \sqrt{x}+\sqrt{y} \right)\times \left( \sqrt{x}-\sqrt{y} \right)=x-y,\left( x\ge 0\,and\,y\ge 0 \right)\]
- \[{{\left( \sqrt{x}+\sqrt{y} \right)}^{2}}x+y+2\sqrt{xy},\left( x\ge 0\,and\,y\ge 0 \right)\]
- \[{{\left( \sqrt{x}-\sqrt{y} \right)}^{2}}x+y-2\sqrt{xy},\left( x\ge 0\,and\,y\ge 0 \right)\]
- \[\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}},\left( x\ge 0\,and\,y\ge 0 \right)\]
- \[\left( a+\sqrt{b} \right)\left( a-\sqrt{b} \right)={{a}^{2}}-b,(b\ge 0)\]
- \[\left( \sqrt{a}+\sqrt{b} \right)\times \left( \sqrt{a}+\sqrt{b} \right)=\sqrt{ac}+\sqrt{bc}+\sqrt{ad}+\sqrt{bd},\left( a\ge 0,b\ge 0,c\ge 0\,and\,d\ge 0 \right)\]
