Number System, Integers, Fractions & Decimals
1. Number System
Place Value (Indian System)
In the Indian system, we group digits as: ONES, TENS, HUNDREDS, THOUSANDS, TEN THOUSANDS, LAKHS, TEN LAKHS, CRORES.
Example: 3,45,67,890 = Three crore forty-five lakh sixty-seven thousand eight hundred ninety.
Comparing Numbers
- The number with MORE DIGITS is LARGER
- If same number of digits: compare from the LEFT
Roman Numerals
| Roman | Value |
|---|---|
| I | 1 |
| V | 5 |
| X | 10 |
| L | 50 |
| C | 100 |
| D | 500 |
| M | 1,000 |
2. Integers
What Are Integers?
Integers = All whole numbers AND their NEGATIVES: ...-3, -2, -1, 0, 1, 2, 3...
Number Line
Negative numbers to the LEFT of zero. Positive numbers to the RIGHT.
Operations with Integers
- Addition of two negatives: (-3) + (-5) = -8
- Subtraction: 5 - (-3) = 5 + 3 = 8
- Multiplication: (-4) × (-3) = +12 (Negative × Negative = Positive)
- Division: (-12) ÷ 3 = -4
3. Fractions
Types of Fractions
| Type | Example |
|---|---|
| Proper (numerator < denominator) | 3/4 |
| Improper (numerator > denominator) | 7/4 |
| Mixed (whole + proper) | 1¾ |
Operations
- Addition/Subtraction: Convert to equivalent fractions with SAME denominator. Then add/subtract numerators.
- Multiplication: (a/b) × (c/d) = (a×c)/(b×d)
- Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c)
Simplest Form
Divide numerator and denominator by their HCF. 12/16 = 3/4 (divided by 4).
4. Decimals
Place Value After Decimal Point
34.567 → 3 tens + 4 ones + 5 tenths + 6 hundredths + 7 thousandths
Operations
- Addition/Subtraction: Align decimal points. Add/subtract normally.
- Multiplication: Multiply ignoring decimal. Count total decimal places in both numbers. Place decimal in the answer.
- Division: Make the divisor a whole number by moving decimal point.
Conversion
- Fraction to Decimal: Divide numerator by denominator. 3/4 = 0.75
- Decimal to Fraction: Write as fraction with denominator 10, 100, 1000... Simplify. 0.25 = 25/100 = 1/4
