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 ↓


Irrational Numbers


In mathematics, numbers play an important role in many different concepts. One of the fascinating types of numbers that often fascinates students is irrational numbers. Understanding irrational numbers will help you understand the broader topic of number systems. Let's take a journey into the world of irrational numbers and explore their characteristics, examples, and their differences from other types of numbers.

What are irrational numbers?

Irrational numbers are numbers that cannot be expressed as a simple fraction, that is, they cannot be written as a ratio of two integers. In other words, irrational numbers cannot be written in the form a/b where a and b are integers and b is not zero.

The decimal expansion of an irrational number is non-terminating and non-repeating. This means that the decimal sequence goes on forever without repeating any pattern. Let's dig deeper with some familiar examples.

Examples of irrational numbers

1. Pi (π)

The most famous example of an irrational number is pi (π). The pi number represents the ratio of the circumference of any circle to its diameter. The decimal expansion of pi begins at 3.14159 and continues to infinity without repeating.

π = 3.141592653589793238...
Diameter π = c/d

2. Square roots of non-perfect squares

Another common example is the square root of numbers that are not perfect squares. For example, the square root of 2 (√2) is irrational. It cannot be expressed exactly as a fraction, and its decimal form is non-repeating and non-terminating.

√2 = 1.414213562373095048...
√2

3. Euler number (e)

The number e, known as the Euler number, is another irrational number. e is the base of the natural logarithm and is used in exponential growth calculations in mathematics, science, and engineering.

e = 2.718281828459045235...

Properties of irrational numbers

Irrational numbers have special properties that set them apart from other types of numbers. Here are some of the main properties of irrational numbers:

Non-repeating decimal

As defined, the decimal representation of an irrational number never ends, nor does it repeat any pattern. For example, dividing 1 by 3 gives a repeating decimal: 0.33333... which is rational. However, the decimal for √2 is non-repeating.

Cannot be expressed as fractions

Unlike rational numbers, irrational numbers cannot be expressed precisely in fraction form. While rational numbers can be written in the form a/b, irrational numbers cannot be expressed this way.

Irrational numbers are dense on the number line

This property means that between any two rational numbers there is at least one irrational number, and vice versa. This density ensures the continuity of real numbers on the number line.

Differentiating between rational and irrational numbers

Understanding the difference between rational and irrational numbers will deepen your understanding of number systems.

Rational numbers

A rational number can be expressed as a fraction a/b where a and b are integers, and b ≠ 0. Examples of rational numbers include 1, 0.5, 2/3, and -4. Rational numbers have decimal expansions that either terminate (for example, 0.75) or repeat (for example, 0.333...).

Irrational numbers

In contrast, irrational numbers have no repeating or terminating decimal pattern and cannot be expressed exactly as fractions. Examples include π, √2, and e.

Illustrating irrational numbers on the number line

Although irrational numbers cannot be expressed as fractions, they can still be represented on the number line. Consider how one might locate an irrational number such as √2 on the number line:

If we construct a right-angled triangle whose two legs have a length of 1 unit, then according to the Pythagorean theorem the hypotenuse of this triangle will be √2.

1 1 √2

Importance of irrational numbers

Irrational numbers are important because they enable mathematicians and scientists to perform calculations with greater accuracy and precision. They appear naturally in calculations involving geometry, trigonometry, and calculus.

For example, pi (π) is essential for calculating the circumference and area of circles, while Euler's number (e) is important for exponential growth models, as found in compound interest, population growth, and radioactive decay.

Approximation of irrational numbers

While you cannot represent irrational numbers exactly as fractions, they can be approximated to any level of precision desired using decimal expansions. These approximations are often used for practical calculations. For example, pi is often approximated as 3.14 or 22/7 for rough calculations.

Closing thoughts

Irrational numbers are a fascinating and important part of the number system. They exemplify the complexity and beauty of mathematics, allowing us to explore concepts that go far beyond simple counting or basic geometric measurements. Understanding irrational numbers opens up the ability to delve deeper into more advanced mathematical concepts and real-world applications that require precise and complex calculations.

As you progress in your study of mathematics, keep in mind that irrational numbers, although not easily expressible, have a fundamental place in the construction of logical and mathematical structures that allow us to understand and describe the world around us. Irrational numbers, with their non-repeating and non-terminating properties, challenge us to think more deeply about the nature of numbers and the infinite possibilities they present.


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Irrational Numbers

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