EXPONENT AND POWERS
POWER
\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m}}-n\]
\[{{5}^{3}}\div {{5}^{2}}={{5}^{3}}-2\]
FUNDAMENTALS

- Exponential form is nothing but repeated multiplication.

- Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

- (a) When a negative number is raised to an even power the value is always positive.

RATIO, PROPORTION AND UNITARY METHOD
RATIO
The comparison of two quantity of same kind by division is called ratio.
Example: Ratio between Rs. 30 and Rs. 50, but there can be no ratio between Rs. 30 and 50 apples, form of ratio \[=x:y\]
\[x\to \]antecedent
\[y\to \]consequent

- B The ratio of \[x\]to \[y\]
- B \[x\]is to \[y\]
- \[x:y\]
- B \[x\]and \[y\] are called terms of ratio.

- Compound Ratio: Ratio is compound when antecedents are multiplied by respective antecedents and consequents are multiplied by respective consequents.

- Duplicate Ratio: If \[x:y\] is a ratio, then\[{{x}^{2}}:{{y}^{2}}\]is duplicate ratio.

ALGEBRAIC EXPRESSIONS
ALGEBRAIC EXPRESSION

\[2x\] | Expression |

\[2x+y\] | Binomial |

\[2x=14\] | Equation |

- Variable: A symbol which takes various values is known as a variable. Normally it is denoted by x, y, z etc.
- Algebraic expression: A combination of constants and variables connected by some or all of the four fundamental operations \[+,\] \[-,\]\[\times \]and \[\div \] is called an algebraic expression.

- Here \[-5\]is the coefficient of the variable 'x' and 6 is the constant.

ALGEBRA
ALGEBRA
The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
Introduction to Algebra

- Variable: A letter symbol which can take various numerical values is called variable or literal.

- Constant: A symbol which can take a fixed numerical value is called a constant.

- Term: Numericals or literals or their combinations by operation of multiplication are called terms.

- Constant Term: A term of an expression having no literal is called a constant term.

- Algebraic expression: A combination of constant and variable connected by mathematical operations \[(+,\,\,-,\,\,\times ,\,\,\div )\] is called an algebraic expression.

FACTORS AND MULTIPLES
FACTOR:

- One Number is said to be a factor of another when it divides the other exactly.
- If a number 'x' divides another number 'y' exactly, then we say that 'x' is a factor of 'y'.

- A multiple of a number is a number obtained by multiplying it by integers.

- If a number 'x' divides another number 'y' exactly, then we say that 'y' is a multiple of x.

- Factors of a number is finite.
- Multiples of a number is infinite.
- more...

FRACTIONS AND DECIMALS
FUNDAMENTALS
Natural numbers: All counting numbers are called natural numbers.

- It is denoted by N.
- \[N=\left\{ 1,2,3,4,\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \right\}\]

- It is denoted by W.
- \[~W=\left\{ 0,1,2,3,\_\_\_\_\_\_\_\_\_\_ \right\}\]
- Fraction: A part of whole is called fraction.

- \[\frac{x\to \text{Numerator}}{y\to \text{Denominator}}\]

- Decimal fraction: A fraction whose denominator is 10, 100, 1000 etc.......is called a decimal fraction.

- more...

NUMBER SYSTEM
FUNDAMENTALS

- Digits: for representing any number we use ten symbols

- Numeral: A group of digits representing a number is called a numeral.

- Notation: The system of expressing a number in figure or digits is called notation.
- Numeration: The logic of representing a number in words is called numeration.

Periods | Crores |
Lakhs more...
SYMMETRY
SYMMETRY
Symmetry is when something is made up of two parts that are exactly the same and they are facing each other.
Line Symmetry
- A line of symmetry divides a figure into two mirror - image halves.
- The line of Symmetry is imaginary line where you could fold the image and have both halves match exactly.
- Letters and line symmetry
- There are some letters which do not have any line symmetry.
- Rhombus:
- Two lines of symmetry AC and BD
- more...
DATA HANDLING
SYMMETRY
- The word 'data' means information. Its exact dictionary meaning is given facts.
- Statistical data are of two types (i) primary (ii) secondary.
- The number of times an observation occurs in the given data is called the frequency of the observation.
- There are two types of frequency distribution.
- Discrete frequency distribution.
- Continuous frequency distribution.
- In a discrete frequency distribution the cumulative frequency of a particular value of the variable is the total of all the frequencies of the values of the variable which are less than or equal to the particular value.
- A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table.
- Bar graphs
- more...
PRACTICAL GEOMETRY
FUNDAMENTALS
- A ruler protractor and compass are used for constructions.
- Given a line 1 and a point P not on it, a line parallel to 1 can be drawn through the point P, using the idea of 'equal alternate angles' or 'equal corresponding angles'.
- The sum of lengths of any two sides of a triangle is greater than its third side.
- The difference of lengths of any two sides of a triangle is lesser than its third side.
- The sum of angles in a triangle is \[180{}^\circ .\]
- The following cases of congruence of triangles, help us construct the triangle.
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