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 ↓


Real Numbers on the Number Line


In mathematics, it is important to understand where real numbers lie on the number line. It is a fundamental concept for many mathematical operations, calculations, and distinctions between numbers. Real numbers consist of all the numbers on the number line, whether positive or negative, and include zero. In this document, we will explore what real numbers are, how they are organized, and how they are represented on the number line.

1. What are real numbers?

Real numbers are all the numbers you can find on the number line. They include both rational numbers and irrational numbers. Rational numbers are numbers that can be expressed as the fraction a/b, where a and b are integers and b ≠ 0 Examples include 1/2, 4, -3.5, etc.

Irrational numbers are numbers that cannot be written as simple fractions, have non-repeating and non-terminating decimals. Examples are π (pi) and √2 (square root of 2).

2. Visualizing the number line

The number line is a straight line where each point corresponds to a real number. To identify and visualize real numbers more easily, imagine a horizontal line:

-3 -2 -1 0 1 2 3 4

Real numbers can be placed at corresponding locations on this line. Real numbers keep increasing indefinitely in both positive and negative directions.

3. Understanding rational numbers on the number line

To place rational numbers on the number line, we need to understand their fractional values. For example, let's consider the fraction 1/2. Numerically, 1/2 is equal to 0.5. On the number line, it is placed in the position between 0 and 1.

1/2

Similarly, -3/4 = -0.75, which lies between -1 and 0.

-3/4

4. Putting irrational numbers on the number line

Irrational numbers are more challenging to plot accurately because they cannot be represented precisely as fractions. Their decimal expansions go on forever without repeating. Let's plot two common irrational numbers: π and √2 on the number line.

Example 1: Substituting π

The value of π (pi) is approximately 3.14159. On the number line, this number is located slightly ahead of 3.

π

Example 2: Substituting √2

The value of √2 is approximately 1.414. It is found slightly above 1 on the number line.

√2

5. Properties of real numbers

Real numbers have several essential properties:

Commutative property

Real numbers obey the commutative properties for addition and multiplication.

a + b = b + a
a × b = b × a

Associative property

Real numbers also obey associative properties.

(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)

Distributive property

The distributive property connects addition and multiplication.

a × (b + c) = a × b + a × c

6. Special categories of real numbers

The real numbers can be divided into some special groups:

Integers

Integers include all positive whole numbers, negative whole numbers, and zero. They do not include fractions or decimals.

..., -3, -2, -1, 0, 1, 2, 3, ...

Whole numbers

Whole numbers include all natural numbers along with zero.

0, 1, 2, 3, 4, ...

Natural numbers

Natural numbers include all positive numbers starting with 1. They are often used in counting.

1, 2, 3, 4, ...

7. Use of real numbers in real life

Real numbers are the basis of our lives. Here are some applications:

Finance

Real numbers are used to represent financial calculations such as interest rates, balances, and profits.

Measurement

Real numbers are used to express quantities in the measurement of everything from length to weight.

Scientific calculations

Real numbers are used in all scientific calculations, from calculating chemical concentrations to physics experiments.

8. Examples and practice problems

Let us try to solve some problems related to real numbers on the number line.

Problem 1: Identifying numbers

Mark the following numbers on the number line: -2, 0.75, √5.

Solution

-2 is directly above and to the left of 0 The decimal 0.75 sits between 0 and 1 Finally, √5 (about 2.236) will sit slightly further away from 2.

Problem 2: Distance on the number line

What is the distance between -3 and 2 on the number line?

Solution

To find the distance, subtract the smaller number from the larger number: 2 - (-3) = 2 + 3 = 5 The distance is 5 units.

9. Conclusion

The concept of real numbers and their place on the number line is extremely important in mathematics. It helps to understand the fundamental nature of numbers, their interactions, and their applicability in solving complex problems. With practice and understanding, real numbers can be a simple, yet powerful tool in your mathematical arsenal.


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Real Numbers on the Number Line

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