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 ↓


Decimal Representation of Real Numbers


In this topic, we will explore how real numbers can be represented in decimal form. Real numbers are a fundamental part of mathematics and appear in many situations in everyday life. Through various examples and illustrations, we will dive into their decimal representation to gain a better understanding.

What are real numbers?

Real numbers include both rational and irrational numbers. These include numbers that can be written as a fraction such as 3/4 or any number that is found on a continuous number line.

What is decimal representation?

Decimal representation refers to expressing numbers using the base ten numeral system. It uses digits from 0 to 9 to represent any real number.

Examples of decimal representation

  • The number three tenths in decimal form is 0.3.
  • One hundred twenty five can be written as 125.0.
  • The fraction 1/2 is converted to decimal as 0.5.

Representing rational numbers as decimals

Rational numbers can be expressed as a ratio or fraction a/b where a and b are integers and b ≠ 0. Their decimal representation can be either terminating or repeating.

End decimals

These are decimal numbers that have a finite number of digits after the decimal point.

Example: 0.75 is the decimal representation of 3/4, which is a rational number. Dividing 3 by 4 gives the terminating decimal.

Repeating decimals

These have a pattern that is repeated indefinitely.

Example: 1/3 has the value 0.333... where the digit '3' is repeated an infinite number of times. We represent it as 0.̅3.

Expressing irrational numbers as decimals

Irrational numbers cannot be expressed as fractions. Their decimal form neither terminates nor repeats.

Example: The decimal value of π (pi) is approximately 3.14159... and expands indefinitely without any pattern.

Visualization of decimal representation

Let's visualize decimal powers of ten:

012

This line shows the numbers in their decimal form. The distance between the dots shows the decimal increments.

Look at the repeating decimals: 0.̅3

00.30.330.333

Converting fractions to decimals

The process of converting fractions to decimals involves division: the numerator is divided by the denominator.

Example:

1/4 = 0.25
2 ÷ 5 = 0.4

Decimal operations

It is very important to understand how to work with decimals. Here is a brief overview of the basic operations:

Addition and subtraction

Align the decimal points vertically and then add or subtract.

12.5
+ 3.75
,
16.25

Multiplication

Multiply as normal, ignoring the decimal points, then place the decimal in the product based on the total number of decimal places in the factors.

0.6
× 0.2
,
0.12

Division

Move the decimal point forward to make the divisor a whole number, then divide as normal.

4.2 ÷ 0.6 = 7

Conclusion

Decimal representation is a fundamental aspect of mathematics that converts complex numerical ideas into an understandable format. It lays the groundwork for further mathematical education and practical application in everyday situations.

Understanding decimals not only empowers mathematical calculations but also enhances logical thinking abilities. Through rigorous practice and exploration, decimal representation can be mastered.


Grade 9 → 1.7


U
username
0%
completed in Grade 9

← Prev (1.6)
Representation on the Number Line
Next (2) →
Polynomials

Comments 



Decimal Representation of Real Numbers

https://www.buddymath.com/g9/1.7?bmal=en