Top Math Topics for Sindh University Entry Test: Numbers, Rational/Irrational, & Quick Sum Tricks

Question: What are Numbers?

Answer: Numbers are the foundation of mathematics, helping us count, measure, compare, and label objects or quantities.

🔹 In simple terms:
Numbers tell us “how many,” “how much,” or “in what position” something is.

📚 Types of Numbers (With Examples)

1️⃣ Natural Numbers (Counting Numbers)

• Used for basic counting.
• Examples: 1, 2, 3, 4, 5, …
• Note: Starts from 1 (does not include 0).

2️⃣ Whole Numbers

• Natural numbers plus zero.
• Examples: 0, 1, 2, 3, 4, …

3️⃣ Integers

• Whole numbers including negatives.
• Examples: -3, -2, -1, 0, 1, 2, 3, …

4️⃣ Rational Numbers

• Can be written as fractions (p/q) where q ≠ 0.
• Examples: ½, -¾, 5 (as 5/1), 0.25 (¼).

5️⃣ Irrational Numbers

• Cannot be written as fractions; decimals never end or repeat.
• Examples: π (pi), √2, √5.

6️⃣ Real Numbers

• All rational and irrational numbers combined.
• Examples: -5, 0, ½, √3, π.

7️⃣ Imaginary Numbers

• Involve the square root of negative numbers.
• Example: √-1 = *i* (where *i* is the imaginary unit).

8️⃣ Complex Numbers

• Combine real and imaginary numbers.
• Example: 4 + 3*i*.

9️⃣ Prime Numbers

• Have only two factors: 1 and itself.
• Examples: 2, 3, 5, 7, 11.

🔟 Composite Numbers

• Have more than two factors.
• Examples: 4, 6, 8, 9, 10.

Rational vs Irrational Numbers: Key Differences Explained

🔹 What is a Rational Number?

A rational number is any number that can be expressed as a fraction (p/q), where:
✔ p and q are integers
✔ q ≠ 0 (denominator cannot be zero)

📌 Examples of Rational Numbers:

• Fractions: 12,3421​,43​
• Whole numbers: 5=51,−3=−315=15​,−3=1−3​
• Terminating decimals: 0.75=340.75=43​
• Repeating decimals: 0.3‾=130.3=31​

✅ Key Properties of Rational Numbers:

✔ Can be positive or negative
✔ Decimal form either terminates or repeats

🔹 What is an Irrational Number?

An irrational number cannot be written as a fraction (p/q).

📌 Examples of Irrational Numbers:

• π (Pi) ≈ 3.14159265… (never ends, no repeating pattern)
• √2 ≈ 1.4142135… (non-terminating, non-repeating)
• √3, √5, e (Euler’s number)

❌ Key Properties of Irrational Numbers:

✖ Cannot be written as p/q
✖ Decimal never ends (non-terminating)
✖ Decimal never repeats (non-repeating)

🔁 Rational vs Irrational Numbers: Quick Comparison

Feature	Rational Numbers	Irrational Numbers
Can be a fraction (p/q)?	✅ Yes	❌ No
Decimal terminates?	✅ Sometimes	❌ Never
Decimal repeats?	✅ Sometimes	❌ Never
Examples	12,5,0.75,0.3‾21​,5,0.75,0.3	π, √2, √3, e

🎯 Key Takeaways

✔ Rational Numbers = Can be written as fractions (terminating or repeating decimals).
✔ Irrational Numbers = Cannot be written as fractions (non-terminating, non-repeating decimals).

🔍 Why Does This Matter?

Understanding the difference helps in algebra, geometry, and advanced math. Rational numbers are common in daily life, while irrational numbers appear in calculations involving π, square roots, and logarithms.

Complete Guide to Rational & Irrational Numbers with Formulas & MCQs

📌 Rational vs Irrational Numbers: Key Differences

✅ Rational Numbers (Q)

• Can be expressed as fractions p/q (q ≠ 0)
• Decimal terminates (0.5) or repeats (0.333…)
• Examples:
  ✔ 1/2 = 0.5 (terminating)
  ✔ 2/3 = 0.666… (repeating)
  ✔ 5 = 5/1 (whole number)

❌ Irrational Numbers

• Cannot be written as p/q
• Decimal never ends & never repeats
• Examples:
  ✖ π = 3.141592…
  ✖ √2 = 1.414213…
  ✖ e (Euler’s number)

🔁 Quick Comparison Table

Feature	Rational	Irrational
Fraction form (p/q)?	✅ Yes	❌ No
Decimal ends?	✅ Sometimes	❌ Never
Decimal repeats?	✅ Sometimes	❌ Never
Examples	1/2, 0.75, 5	π, √3, e

🧮 Important Number Series Formulas

1. Sum of First ‘n’ Natural Numbers

📌 Formula:Sum=n(n+1)2Sum=2n(n+1)​

Examples:
✔ Sum of 1 to 10 = 55
✔ Sum of 1 to 100 = 5050

2. Sum of First ‘n’ Even Numbers

📌 Formula:Sum=n(n+1)Sum=n(n+1)

Example:
Sum of first 5 even numbers (2+4+6+8+10) = 30

3. Sum of First ‘n’ Odd Numbers

📌 Formula:Sum=n2Sum=n2

Example:
Sum of first 5 odd numbers (1+3+5+7+9) = 25

📝 Practice MCQs (With Explanations)

1️⃣ Which is a natural number?
A) -1
B) 0
C) 5
✅ Answer: C (Natural numbers start from 1)

2️⃣ Which is NOT a whole number?
A) 0
B) 10
C) -5
✅ Answer: C (Whole numbers ≥ 0)

3️⃣ Which is rational?
A) √2
B) π
C) 1/3
✅ Answer: C (Can be written as p/q)

4️⃣ What type of number is √25?
A) Irrational
B) Rational
C) Natural
✅ Answer: D) Both B & C (√25 = 5)

5️⃣ Sum of rational + irrational = ?
A) Always Rational
B) Always Irrational
✅ Answer: B (Rational + Irrational = Irrational)

6️⃣ Which is a terminating decimal?
A) √3
B) 5/8
✅ Answer: B (5/8 = 0.625)

7. If *a* is rational and *b* is irrational, then a + b is:

A) Always rational
B) Always irrational
C) Can be rational or irrational
D) An integer

✅ Answer: B (Rational + Irrational = Irrational.)

8. The product of a non-zero rational and an irrational number is:

A) Always rational
B) Always irrational
C) Sometimes rational
D) An integer

✅ Answer: B (Example: 2 × √3 = 2√3 → irrational.)

9. Which of the following is rational?

A) (√2 + √3)²
B) √16/√9
C) π + 2
D) √5 × √5

✅ Answer: B (√16/√9 = 4/3, a fraction → rational.)

10. The decimal expansion of √2 is:

A) Terminating
B) Non-terminating but repeating
C) Non-terminating & non-repeating
D) Finite

✅ Answer: C (√2 ≈ 1.414213…, non-repeating & infinite.)

11. What is the sum of first 10 natural numbers?

A) 45
B) 50
C) 55
D) 60

✅ Answer: C
📌 Explanation: Using formula n(n+1)/2 = 10×11/2 = 55.

12. The sum of two consecutive even numbers is 34. What are the numbers?

A) 14, 16
B) 16, 18
C) 18, 20
D) 20, 22

✅ Answer: B
📌 Explanation: Let numbers be x and x+2 → x + (x+2) = 34 → x=16 → 16,18.

13. What is 1+2+3+…+50?

A) 1250
B) 1275
C) 1300
D) 1350

✅ Answer: B
📌 Explanation: Sum of first n numbers = n(n+1)/2 → 50×51/2 = 1275.

14. The sum of first ‘n’ odd numbers is 225. What is ‘n’?

A) 13
B) 15
C) 17
D) 19

✅ Answer: B
📌 Explanation: Sum of first n odd numbers = n² → n²=225 → n=15.

15. What is the sum of first 20 even numbers?

A) 400
B) 410
C) 420
D) 440

✅ Answer: C
📌 Explanation: Sum = n(n+1) → 20×21 = 420.

🎯 Key Takeaways

✔ Rational Numbers = Fractions (p/q), terminating/repeating decimals
✔ Irrational Numbers = Non-terminating, non-repeating (π, √2)
✔ Sum Formulas:

• Natural Numbers: n(n+1)22n(n+1)​
• Even Numbers: n(n+1)n(n+1)
• Odd Numbers: n2n2