Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Number Systems ↓


Operations on Real Numbers


Real numbers can be understood as numbers that can be found on the number line. These include both rational numbers (such as fractions, finite and repeating decimals) and irrational numbers (such as the square root of 2 and non-repeating, non-terminating decimals). In mathematics, it is important to understand operations on real numbers as they form the basis of algebra and other branches of mathematics.

1. Addition of real numbers

Addition is one of the basic operations in mathematics. When you add two real numbers, the result is also a real number.

1.1 Adding positive numbers

If you add two positive numbers, you simply move to the right on the number line.

3 + 5 = 8
0 3 8

The number line shows going from 3 to 8 by moving 5 units to the right.

1.2 Adding negative numbers

If you add two negative numbers, you move to the left on the number line.

(-3) + (-5) = -8
-5 -3 -8

The above number line shows -8 when moving 3 and 5 units to the left from zero.

1.3 Adding numbers with different signs

When you add a positive and a negative number, you find the difference and move in the direction of the maximum absolute value.

3 + (-5) = -2
3 -5 -2

The number line above shows starting from 3 and moving from zero to the left till -2.

2. Subtraction of real numbers

Subtraction can be thought of as the opposite operation of addition. Subtracting a real number means adding its opposite.

2.1 Subtracting positive numbers

To subtract a positive number, you move left on the number line.

5 - 3 = 2
5 2 0

You start at 5 and move 3 units to the left and stop at 2.

2.2 Subtracting negative numbers

Subtracting a negative number is equivalent to adding the corresponding positive number.

5 - (-3) = 8
5 8

Starting from 5 and moving 3 units to the right, we reach 8.

3. Multiplication of real numbers

Multiplication of real numbers is a repeated addition. The product of two real numbers always gives a real number.

3.1 Multiplication of positive numbers

When you multiply two positive numbers, the result is positive.

3 × 5 = 15
15

Multiply 3 and 5 to get 15, moving to the right as many times as necessary to calculate it.

3.2 Multiplying a positive and a negative number

The product of a positive and a negative number is negative.

3 × (-5) = -15
0 -15

Here the opposite direction on the left side is considered to show the relation through multiplication.

3.3 Multiplication of two negative numbers

The product of two negative numbers is positive.

(-3) × (-5) = 15
15

Here, the corollary involves realizing that reversing direction still yields the same final moves.

4. Division of real numbers

Division is a process that is based on dividing a number into equal parts. There are several rules to remember when dividing real numbers, especially negative numbers.

4.1 Division of positive numbers

When a positive number is divided by another positive number, the result is positive.

15 ÷ 3 = 5
15 5

The grouping similarly shows the millet strips resulting in cumulative allocation as explained above.

4.2 Division of a positive and a negative number

Dividing a positive number by a negative number gives a negative result, and vice versa.

15 ÷ (-3) = -5
-15 -5

This number is summed up to get the opposite corners and gives the result.

4.3 Division of two negative numbers

When two negative numbers are divided, the result is positive.

(-15) ÷ (-3) = 5
15 5

Both negativity ends and everything becomes positive.

5. Conclusion

Understanding operations on real numbers can be very useful in mathematics and everyday life. These operations follow specific, predictable patterns and rules. Learning these mathematical concepts becomes more intuitive and meaningful through practical exercises and clear visualizations, such as number lines and examples.


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Operations on Real Numbers

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