Whole Numbers
Successor of a Number
n + 1
The successor of a number is the next number in the sequence.
Predecessor of a Number
n – 1
The predecessor of a number is the previous number in the sequence.
Properties of Whole Numbers
Commutative Property:
Addition: a + b = b + a
Multiplication: a × b = b × a
The order of numbers does not affect the sum or product.
Associative Property:
Addition: (a + b) + c = a + (b + c)
Multiplication: (a × b) × c = a × (b × c)
Grouping of numbers does not affect the sum or product.
Distributive Property:
a × (b + c) = a × b + a × c
Distributes multiplication over addition.
Playing with Numbers
Factors and Multiples
Factors are numbers that divide another number exactly, while multiples are the product of a number and an integer.
Prime and Composite Numbers
Prime numbers have exactly two factors (1 and itself), while composite numbers have more than two factors.
Divisibility Rules
Rules that help determine if a number is divisible by another without actual division.
HCF (Highest Common Factor)
The largest number that divides two or more numbers without a remainder.
LCM (Least Common Multiple)
The smallest number that is a multiple of two or more numbers.
Basic Geometrical Ideas
Perimeter of a Rectangle
2 × (Length + Breadth)
The perimeter is the total distance around the rectangle.
Perimeter of a Square
4 × Side
The perimeter is the total distance around the square.
Understanding Elementary Shapes
Types of Angles
Acute: Less than 90 degrees
Right: Exactly 90 degrees
Obtuse: Between 90 and 180 degrees
Straight: Exactly 180 degrees
Reflex: Between 180 and 360 degrees
Complete: Exactly 360 degrees
Classification of Triangles
Based on sides:
Scalene: All sides are different
Isosceles: Two sides are equal
Equilateral: All sides are equal
Based on angles:
Acute-angled: All angles are less than 90 degrees
Right-angled: One angle is 90 degrees
Obtuse-angled: One angle is more than 90 degrees
Integers
Addition of Integers
Examples:
1. 5 + 3 = 8 (Both integers are positive, so we add them.)
2. -5 + (-3) = -8 (Both integers are negative, so we add their absolute values and keep the negative sign.)
3. 5 + (-3) = 2 (Integers have different signs, so we subtract 3 from 5 and keep the positive sign of 5.)
4. -5 + 3 = -2 (Integers have different signs, so we subtract)
Fractions
Fraction Addition
a/b + c/b = (a + c)/b
Add fractions with the same denominator by adding the numerators.
Fraction Subtraction
a/b – c/b = (a – c)/b
Subtract fractions with the same denominator by subtracting the numerators.
Equivalent Fractions
Fractions that represent the same value when simplified.
Simplifying Fractions
Reducing fractions to their simplest form by dividing the numerator and denominator by their GCD.
Data Handling
Mean (Average)
Formula:
Mean = Sum of observations / Number of observations
Example:
Consider the data set: 4, 8, 6, 5, 3
Step-by-Step Calculation:
1. Add up all the values: 4 + 8 + 6 + 5 + 3 = 26
2. Count the number of values: There are 5 values in the data set.
3. Divide the sum by the number of values: Mean = 26 / 5 = 5.2
So, the mean (average) of the data set 4, 8, 6, 5, 3 is 5.2.
Mensuration
Perimeter of a Rectangle
2 × (Length + Breadth)
The perimeter is the total distance around the rectangle.
Perimeter of a Square
4 × Side
The perimeter is the total distance around the square.
Area of a Rectangle
Length × Breadth
The area is the amount of space inside the rectangle.
Area of a Square
Side × Side
The area is the amount of space inside the square.
Algebra
Simple Equations
Equations that involve finding the value of a variable that makes the equation true.
Ratio and Proportion
Ratio
Ratio of a to b = a/b
A ratio compares two quantities.
Proportion
a/b = c/d implies a × d = b × c
Proportion states that two ratios are equal.
