🗺️ Chapter Roadmap
1 Types of Numbers
The world of mathematics is built upon numbers. In competitive exams, direct MCQs are often asked about their definitions. Let's understand them in simple terms:
Natural Numbers
These are counting numbers. We always start counting things from 1.
Whole Numbers
If we include Zero (0) in the set of Natural numbers, they become Whole numbers.
Integers
These include Positive (+), Negative (-) numbers, and Zero (0). (But they do not include fractions or decimal point numbers).
Rational Numbers
Any number that we can write in the form of a pq (Fraction). The condition is that the bottom part (q) must not be zero.
Examples: 12, -34, 5 (because 5 = 51)
Irrational Numbers
Numbers that CANNOT be written in the form of a pq fraction. Their decimal never ends and never repeats.
Famous Examples:
2 Even, Odd & Prime Numbers
Even Numbers
Numbers that are perfectly divisible by 2 (can be put in pairs).
Odd Numbers
Numbers that are not divisible by 2 (cannot be put in pairs).
⭐ Prime vs Composite (Exam Favorite)
Prime Numbers
Numbers that only come in the table of 1 and Themselves. They have exactly 2 factors.
Composite Numbers
Numbers that have more than 2 factors (meaning, they come in other tables besides their own).
- The number '1' is neither Prime nor Composite. It is a unique number.
- The number '2' is the only Even Prime number. All other prime numbers are Odd.
- Twin Primes: Two prime numbers that have a difference of 2. Example: (3, 5), (5, 7), (11, 13).
3 LCM and HCF (Basic Concept)
Knowing LCM is very important for adding/subtracting fractions.
HCF (Highest Common Factor)
This is the largest number that perfectly divides all the given numbers.
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
HCF = 6 (Largest common)
LCM (Least Common Multiple)
This is the smallest number that comes in the table of all the given numbers.
Multiples of 4 = 4, 8, 12, 16, 20, 24...
Multiples of 6 = 6, 12, 18, 24, 30...
LCM = 12 (First/smallest common)
4 Divisibility Rules (Magic Tricks)
To calculate faster in exams, it is essential to remember these shortcuts. With these, you can instantly tell what a large number is divisible by.
| Rule | Trick (Identification) | Mathematical Example |
|---|---|---|
| 2 | If the last digit of the number is Even (0, 2, 4, 6, 8) | 1548 ✅ |
| 3 | Add all the digits. If the total is divisible by 3. | 342 → (3+4+2=9) ✅ |
| 4 | If the last two (2) digits come in the table of 4. | 512 → (12 ÷ 4 = 3) ✅ |
| 5 | The last digit is either 0 or 5. | 2345, 100 ✅ |
| 6 | A number that passes the rules for both 2 and 3. | 18 → Even & (1+8=9) ✅ |
| 9 | Like the rule of 3: The total of all digits is divisible by 9. | 729 → (7+2+9=18) ✅ |
| 11 | Add alternate digits. Then find their difference. The answer should be 0 or a multiple of 11. | 1331 → (1+3)-(3+1)=0 ✅ |
5 The BODMAS Rule (Order of Operations)
In math, if there is a long equation containing +, -, ×, ÷ all together, we cannot solve it however we want. We must follow this strict order:
📝 Basic Example:
-
1
B (Brackets): First, solve the bracket (4 - 2).
= 20 + 15 ÷ 3 × 2 -
2
D (Division): Next, do the division (15 ÷ 3).
= 20 + 5 × 2 -
3
M (Multiplication): Then, do the multiplication (5 × 2).
= 20 + 10 -
✓
A (Addition): Finally, do the addition (20 + 10).
= 30
🧠 Advanced Example (Nested Brackets):
If there is more than one bracket, always solve the smallest bracket ( ) first, then the curly bracket { }, and finally the square bracket [ ].
1. Small Brackets (8 - 6) = 2 = 18 - [ 6 - { 4 - 2 } ]
2. Curly Brackets {4 - 2} = 2 = 18 - [ 6 - 2 ]
3. Square Brackets [6 - 2] = 4 = 18 - 4
4. Final Subtraction = 14