What are Integers?
Integers are whole numbers that can be positive, negative, or zero. They include all positive numbers \(\{1, 2, 3, ...\}\), all negative numbers \(\{-1, -2, -3, ...\}\), and zero \(\{0\}\). Integers do NOT include fractions or decimals.
The Integer Family:
- Positive integers: \(1, 2, 3, 4, ...\) (numbers to the right of zero on number line)
- Negative integers: \(-1, -2, -3, -4, ...\) (numbers to the left of zero on number line)
- Zero: \(0\) (neither positive nor negative)
Operations on Integers
| Operation | Rule | Example |
|---|---|---|
| Addition | Same signs: Add and keep the sign Different signs: Subtract and use sign of larger number | \((-5) + (-3) = -8\) \((-7) + 4 = -3\) |
| Subtraction | Add the opposite: \(a - b = a + (-b)\) | \(5 - 8 = 5 + (-8) = -3\) |
| Multiplication | Same signs = Positive product Different signs = Negative product | \((-4) \times (-2) = 8\) \((-3) \times 5 = -15\) |
| Division | Same signs = Positive quotient Different signs = Negative quotient | \((-12) \div (-3) = 4\) \(10 \div (-2) = -5\) |
Real-life Example:
- Temperature: If it's \(-5^\circ\)C and drops another \(7^\circ\)C, the new temperature is \(-12^\circ\)C (addition of negatives)
- Bank balance: If you have ₹500 and spend ₹700, your balance becomes ₹-200 (negative integer)
Rules for Adding Integers (Number Line Method):
- Start at the first number
- Move right for positive numbers, left for negative numbers
- Where you land is your answer
Rules for Subtracting Integers (Keep-Change-Change):
- Keep the first number
- Change subtraction to addition
- Change the sign of the second number