Real Numbers: Types, Properties & Operations

TAKEAWAYS:

• Real numbers include rational (fractions/terminating decimals) and irrational (non-terminating, non-repeating decimals) numbers.
• Key properties of real numbers are commutative, associative, distributive, identity, and inverse.
• Operations on real numbers follow rules like BODMAS and sign rules for addition/subtraction.
• Decimal expansions help classify numbers (terminating, repeating, or non-repeating).
• Real numbers are represented on a number line and used in everyday calculations.

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1. Types of Real Numbers

Real numbers are divided into two main types:

• Rational Numbers (Q): Can be written as fractions $\frac{p}{q}$ where $p$ and $q$ are integers and $q\ne 0$.

  • Examples: $2,\frac{3}{4},0.75,-1.5$
  • Terminating decimals: End after a finite number of digits (e.g., $0.5=\frac{1}{2}$).
  • Repeating decimals: Have repeating patterns (e.g., $0.\overline{3}=\frac{1}{3}$).

• Irrational Numbers (Q'): Cannot be written as fractions; their decimal expansions are non-terminating and non-repeating.

  • Examples: $\sqrt{2},\pi ,e$

Example 1: Classify the following numbers:

• $0.375$ → Terminating decimal → Rational ($\frac{3}{8}$)
• $\sqrt{5}$ → Non-terminating, non-repeating → Irrational
• $0.\overline{6}$ → Repeating decimal → Rational ($\frac{2}{3}$)

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2. Properties of Real Numbers

Real numbers follow four key properties:

Example 2: Verify the distributive property for $a=2,b=3,c=4$: $$a(b+c)=2(3+4)=2\times 7=14$$ $$ab+ac=(2\times 3)+(2\times 4)=6+8=14$$ Both sides are equal, confirming the property.

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3. Operations on Real Numbers

A. Addition and Subtraction

• Same signs: Add absolute values, keep the sign.
  • $5+3=8$
  • $-2+(-4)=-6$
• Different signs: Subtract absolute values, take the sign of the larger number.
  • $7+(-5)=2$
  • $-6+8=2$

B. Multiplication and Division

• Same signs: Result is positive.
  • $3\times 4=12$
  • $(-2)\times (-5)=10$
• Different signs: Result is negative.
  • $6\times (-3)=-18$
  • $(-8)÷2=-4$

Example 3: Solve $(-3)\times 4+(-2)\times (-5)$:

1. Multiply: $(-3)\times 4=-12$, $(-2)\times (-5)=10$
2. Add: $-12+10=-2$

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4. Decimal Expansions

• Terminating decimals: End after a few digits (e.g., $0.5,0.75$).
• Repeating decimals: Have a repeating pattern (e.g., $0.\overline{3},0.1\overline{6}$).
• Non-repeating decimals: Irrational numbers (e.g., $\pi ,\sqrt{2}$).

Example 4: Convert $0.\overline{142857}$ to a fraction: Let $x=0.\overline{142857}$. Multiply by $10^6$ (since the repeating part has 6 digits): $$1,000,000x=142,857.\overline{142857}$$ Subtract the original $x$: $$999,999x=142,857⟹x=\frac{142,857}{999,999}=\frac{1}{7}$$

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5. Number Line Representation

Real numbers can be plotted on a number line, where:

• Positive numbers are to the right of 0.
• Negative numbers are to the left of 0.
• Zero is the origin.

Example 5: Locate $-1.5,0,\frac{3}{2},-\sqrt{4}$ on the number line.

• $-1.5$ → Between -2 and -1
• $0$ → Origin
• $\frac{3}{2}=1.5$ → Between 1 and 2
• $-\sqrt{4}=-2$ → At -2

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6. Applications of Real Numbers

Real numbers are used in:

• Finance: Calculating interest, budgets, and profits.
• Science: Measuring temperature, distance, and time.
• Engineering: Designing structures and circuits.
• Daily life: Cooking measurements, sports scores, and travel distances.

Example 6: A shopkeeper sells apples at ₹12.50 each. If he sells 8 apples, how much money does he earn? $$\text{Total earnings}=12.50\times 8=₹100$$

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Exam Tip

1. Memorize properties: Commutative, associative, and distributive properties are frequently tested.
2. Classify numbers correctly: Know the difference between rational and irrational numbers.
3. Practice decimal conversions: Be able to convert repeating decimals to fractions.
4. Apply BODMAS rules: Always follow the correct order of operations.
5. Number line questions: Expect questions on locating numbers or finding distances between points.

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NEB Board-Style Questions

1. Short Answer:

   • Classify $\sqrt{9},\mathrm{0.333...},\pi$ as rational or irrational.
   • Verify the associative property for $a=1,b=2,c=3$ in addition.

2. Long Answer:

   • Solve: $(-5)\times 3+(-2)\times (-4)$. Show each step.
   • Convert $0.\overline{27}$ to a fraction.

3. Application:

   • A car travels 150 km in 3 hours. What is its average speed in km/h?
   • If $x=-2$, evaluate $3x^2-4x+5$.

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Key Takeaway: Mastering real numbers is essential for higher math topics like algebra, calculus, and statistics. Practice daily to build confidence!