Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Number Systems ↓


Rational Numbers


In the world of mathematics, numbers are classified into different types to help us understand and perform calculations efficiently. One important category is rational numbers. But what exactly are rational numbers? Let's understand this concept in depth so that by the end the concept becomes clear and understandable.

Understanding what rational numbers are

Rational numbers are numbers that can be expressed as a fraction a/b, where a and b are integers, and b (the denominator) is not equal to zero. Here, a is called the numerator, and b is called the denominator. The word "rational" is derived from the word "ratio", which in this context means the ratio of two integers.

Examples of rational numbers include:

  • 1/2 (a simple fraction)
  • -3/4 (a negative fraction)
  • 6/1 (integer expressed as a fraction)
  • 0 (can be written as 0/1)

Visual representation of rational numbers

-1/2 1/2

In the image above, the black line represents the number line with rational numbers. The red point corresponds to -1/2, and the blue point corresponds to 1/2. These points show that rational numbers can be placed precisely on the number line.

Properties of rational numbers

Closing assets

For addition, subtraction, multiplication, and division (except by zero), rational numbers are closed. This means that when you perform any of these operations on two rational numbers, you will always get another rational number as the result.

(3/4) + (1/2) = (3*2 + 4*1) / (4*2) = 10/8 = 5/4

Commutative and associative properties

Rational numbers obey commutative and associative properties for addition and multiplication. This means:

(a/b) + (c/d) = (c/d) + (a/b) [(a/b) + (c/d)] + (e/f) = (a/b) + [(c/d) + (e/f)]

Distributive property

Rational numbers also follow the distributive property as shown below:

(a/b) * [(c/d) + (e/f)] = (a/b) * (c/d) + (a/b) * (e/f)

Existence of additive and multiplicative inverses

For any rational number a/b there exists an additive inverse -a/b such that:

(a/b) + (-a/b) = 0

Similarly, every nonzero rational number a/b has a multiplicative inverse b/a such that:

(a/b) * (b/a) = 1

Conversion between forms

From decimal to fraction

A decimal number is rational if it can be expressed as a fraction. There are two types of rational decimals: terminating and repeating.

For the ending decimal, you can convert it directly into a fraction by using the number of decimal places as a power of ten for your denominator. For example, consider the decimal number 0.75.

0.75 = 75/100 = 3/4

For repeating decimals

Consider the repeating decimal 0.666... as a rational number. Let x = 0.666... then multiply by 10, which gives:

10x = 6.666...

Subtracting these equations, 9x = 6, which gives:

x = 6/9 = 2/3

Comparing rational numbers

When comparing two rational numbers a/b and c/d, you can cross-multiply for simplicity:

a/b > c/d if and only if ad > bc

Consider 1/3 and 2/5, and determine which is larger:

1*5 = 5 and 3*2 = 6, thus 1/3 < 2/5

Operations on rational numbers

Addition and subtraction

To add or subtract rational numbers, you must have the same denominator. For example:

(1/3) + (2/5) = (1*5 + 2*3) / 15 = 5/15 + 6/15 = 11/15

Multiplication

Multiply the numerators together and the denominators together:

(1/3) * (2/5) = (1*2) / (3*5) = 2/15

Division

Multiply by the reciprocal of the divisor:

(1/3) ÷ (2/5) = (1/3) * (5/2) = 5/6

Why study rational numbers?

Rational numbers play an important role in various mathematical concepts and real-life scenarios. They are fundamental in fractions, ratios, and proportions used in various fields such as physics, engineering, and economics. Understanding rational numbers helps in understanding advanced mathematical theories including algebra and calculus.

Conclusion

Rational numbers are an integral part of the number system. Recognizing their properties and knowing how they interact with other numbers can significantly enhance mathematical understanding. Whether representing simple fractions or complex ratios, rational numbers represent a wide range of possibilities in mathematical topics.


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Rational Numbers

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