What is Real Numbers Class 10: Definition & Key Concepts

Definition of Real Numbers in Class 10 Mathematics

Real numbers are the set of all numbers that can be found on the number line. This set includes both rational numbers (numbers that can be expressed as a fraction $rac{p}{q}$ where $p$ and $q$ are integers and $q eq 0$) and irrational numbers (numbers that cannot be expressed as a simple fraction).

In simpler terms, real numbers include:

• Natural numbers (1, 2, 3, ...)
• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers (fractions and decimals that terminate or repeat)
• Irrational numbers (non-terminating, non-repeating decimals like $\sqrt{2}$, $\pi$)

Real numbers form the foundation of many mathematical concepts and are a key part of the Class 10 NCERT syllabus.

Types of Real Numbers Explained

Real numbers are broadly classified into two categories:

1. Rational Numbers:

• Can be written as a ratio of two integers.
• Their decimal expansion either terminates (e.g., 0.75) or repeats periodically (e.g., 0.333...).

2. Irrational Numbers:

• Cannot be expressed as a ratio of two integers.
• Their decimal expansion is non-terminating and non-repeating.
• Examples include $\sqrt{3}$, $\pi$, and $e$.

Understanding these types helps in solving problems related to real numbers in Class 10 exams.

Properties of Real Numbers You Must Know

Real numbers follow several important properties that make calculations easier:

• Closure Property: The sum, difference, or product of any two real numbers is always a real number.
• Commutative Property: $a + b = b + a$ and $ab = ba$ for any real numbers $a$ and $b$.
• Associative Property: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.
• Distributive Property: $a(b + c) = ab + ac$.

These properties are fundamental in algebra and are frequently tested in Class 10 exams.

Worked Example:

If $a = 3.5$ and $b = \sqrt{2}$, verify the distributive property for $a(b + 2)$.

Calculate:

$$a(b + 2) = 3.5(\sqrt{2} + 2) = 3.5 \times \sqrt{2} + 3.5 \times 2 = 3.5\sqrt{2} + 7$$

Calculate separately:

$$ab + 2a = 3.5 \times \sqrt{2} + 2 \times 3.5 = 3.5\sqrt{2} + 7$$

Both expressions are equal, confirming the property.

Euclid’s Division Lemma and Its Role in Real Numbers

Euclid’s Division Lemma is a key concept used in the Real Numbers chapter to prove important results like the fundamental theorem of arithmetic.

Statement: For any two positive integers $a$ and $b$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that:

$$a = bq + r, \quad 0 \leq r < b$$

This lemma helps in:

• Finding the highest common factor (HCF) of two numbers.
• Proving that every composite number can be expressed as a product of primes uniquely.

Example:

Find $q$ and $r$ when $a=23$ and $b=5$.

Divide 23 by 5:

$$23 = 5 \times 4 + 3$$

So, $q=4$ and $r=3$.

Euclid’s lemma is fundamental for understanding divisibility and factorization in Class 10 Maths.

Representation of Real Numbers on the Number Line

One of the key ways to understand real numbers is by representing them on a number line.

• Every real number corresponds to a unique point on the number line.
• Rational numbers are represented by points with exact fractional or decimal values.
• Irrational numbers are located approximately but precisely on the number line, for example, $\sqrt{2}$ lies between 1.4 and 1.5.

Steps to represent $\sqrt{2}$ on the number line:

1. Draw a unit length $OA = 1$ on the number line. 2. At point $A$, draw a perpendicular line and mark $AB = 1$. 3. Join $OB$. 4. $OB$ is the hypotenuse of right triangle $OAB$, so by Pythagoras theorem:

$$OB = \sqrt{OA^2 + AB^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$$

5. Using a compass, mark this length $OB$ on the number line from $O$. This point represents $\sqrt{2}$.

This method helps visualize irrational numbers clearly, which is crucial for Class 10 students.

Decimal Expansions of Real Numbers

Real numbers can be expressed in decimal form, which helps distinguish rational and irrational numbers:

• Terminating decimals: Decimal expansion ends after finite digits (e.g., 0.75).
• Non-terminating repeating decimals: Decimal digits repeat in a pattern (e.g., 0.666...).
• Non-terminating non-repeating decimals: No repeating pattern, representing irrational numbers (e.g., $\pi = 3.14159...$).

Example: Convert $\frac{7}{8}$ to decimal.

Divide 7 by 8:

$$7 \div 8 = 0.875$$

This is a terminating decimal, so $\frac{7}{8}$ is rational.

Understanding decimal expansions is important for identifying number types in Class 10 exams.