Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Number Systems → Real Numbers ↓


Rational and Irrational Numbers


In the field of mathematics, numbers play an essential role. They form the basis for understanding concepts and solving problems that help us navigate the world. The two most important categories within numbers are rational numbers and irrational numbers. Together, they make up the group of real numbers. Understanding these concepts can give you a deeper understanding of how numbers work.

Understanding real numbers

Real numbers are all the numbers you might encounter in everyday life. These include:

  • whole numbers
  • Integers
  • Different
  • Decimal

The real number line is like a vast ocean. Its edges start with the numbers you're familiar with, like zero and one. But as you go further, you find both rational and irrational numbers. Let's dig deeper.

Rational numbers

Rational numbers are numbers that can be expressed as fractions where both the numerator and denominator are integers and the denominator is not zero. In simple terms, if you can write a number in the form a/b, where a and b are integers and b ≠ 0, then it is a rational number.

Examples: 1/2, -3/4, 5, 0.75, -7 Non-example: 3/0 (denominator cannot be zero)

Visual examples of rational numbers

Consider a number line where every point corresponds to a real number. Below is a simplified version that you can visualize:

0 1 2 1/2 3/2

Every point on the number line that corresponds to a fractional value with an integer numerator and a non-zero integer denominator can be considered rational.

Decimals as rational numbers

Rational numbers include all numbers that can be represented in decimal form. If a decimal number has a repeating or terminating sequence, it is rational. This means that it either ends after some number of digits or repeats the same group of digits indefinitely.

Examples of Terminating Decimals: 0.5, 1.25, -3.75 Examples of Repeating Decimals: 0.333..., 2.666..., -12.1212... Non-example (Irrational Number): 0.1010010001... (no repeating pattern)

Irrational numbers

Unlike rational numbers, irrational numbers cannot be written as simple fractions. Their decimal expansion goes on forever without repeating. They represent quantities that cannot be precisely calculated as a ratio of two integers.

Examples of irrational numbers

  • √2 (square root of 2)
  • π (pi)
  • e (Euler number)

These numbers are often found in geometry, calculus, and other advanced areas of mathematics.

Illustrating irrational numbers

Like rational numbers, irrational numbers can also be visualized on the number line. However, they will not land exactly on the fraction-based checkpoint:

√2 π

Note that points such as √2 or π (which lie at about 1.414 and 3.141, respectively) are not equivalent to any simple fractions.

Real-life examples

Rational and irrational numbers are not just theoretical; they have practical applications too.

Rational numbers in life

Things like sharing a cake among friends, measuring a cup of flour, or even the distance covered in a race often result in rational numbers. These are straightforward situations where you can express values as fractions or whole numbers.

Irrational numbers in life

Irrational numbers also have their place. Consider the circumference of circles, which often involves π when calculating the perimeter. Also, diagonal lengths in square areas often result in values involving square roots, such as √2 when trying to find the diagonal of a 1x1 square.

Conversion between forms

There are many instances when you might try to convert a repeating decimal into a fraction or identify an irrational number like √50 into a simpler form.

From decimal to fraction

For example, a recurring decimal 0.666... can be converted into a fraction:

Let x = 0.666... 10x = 6.666... Subtract: 10x - x = 6.666... - 0.666... 9x = 6 x = 6/9 = 2/3

Recognizing simple irrational forms

Knowing that √50 can be simplified:

√50 = √(25 * 2) = √25 * √2 = 5√2

Summary

To summarize, real numbers encompass both rational and irrational numbers. Rational numbers, which can be recognized by their expression as simple fractions and repeating decimals, provide clarity in everyday uses. In contrast, irrational numbers capture values that require infinite precision, where the decimal expansion neither terminates nor repeats. Recognizing the differences and relationships between these types gives you a better understanding of the number system and its infinite beauty.

Exploring these number types gives us not only a better understanding of mathematics, but also a better understanding of the world that mathematics seeks to define and describe.


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Rational and Irrational Numbers

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