Integers are the set of whole numbers along with their negatives: ... -3, -2, -1, 0, 1, 2, 3 ... They extend the natural numbers to include negative values, which are essential for representing temperatures below zero, debt, floors below ground level, and many real-life situations.
Key Concepts
The number line is the best way to visualise integers. Positive integers lie to the right of zero, negative integers to the left. The further left a number is, the smaller it is: -5 < -2 < 0 < 3.
Addition of Integers: When you add two integers with the same sign, add their absolute values and keep the sign. When the signs differ, find the difference of absolute values and use the sign of the number with the larger absolute value.
Subtraction of Integers: Subtracting an integer is the same as adding its opposite. So a - b = a + (-b).
- Multiplication of Integers:
- Positive x Positive = Positive
- Negative x Negative = Positive
- Positive x Negative = Negative
- Negative x Positive = Negative
Division of Integers: The sign rules for division are the same as for multiplication.
- Properties:
- Closure: Integers are closed under addition, subtraction, and multiplication (but NOT division).
- Commutativity: a + b = b + a and a x b = b x a.
- Associativity: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
- Distributivity: a x (b + c) = a x b + a x c.
Find (-7) + 4.
The signs are different. Difference of absolute values = 7 - 4 = 3. The larger absolute value belongs to -7, so the answer is -3.
Find (-5) - (-3).
Change subtraction to addition of opposite: (-5) + 3 = -2.
Find (-6) x (-4).
Both negative, so result is positive: (-6) x (-4) = 24.
Find (-36) / 9.
Different signs, so result is negative: (-36) / 9 = -4.
Simplify (-3) x [(-4) + 2].
First solve bracket: (-4) + 2 = -2. Then (-3) x (-2) = 6.
The temperature in Shimla was -5 degrees C at 6 am. It rose by 8 degrees by noon. What was the temperature at noon?
Temperature at noon = -5 + 8 = 3 degrees C.
A submarine is at a depth of -240 m. It ascends 60 m each hour. After 3 hours, what is its depth?
Depth = -240 + (3 x 60) = -240 + 180 = -60 m.
- Key Formulas:
- a - b = a + (-b)
- (-a) x (-b) = a x b
- a x (-b) = -(a x b)
Common mistakes
Do not confuse the sign of a multiplication result. Remember: two negatives multiplied always give a positive. Also, when solving expressions with brackets, always simplify the bracket first.
Summary
Integers include positive numbers, negative numbers, and zero. They follow clear rules for all four operations, and are closed under addition, subtraction, and multiplication. Understanding sign rules is the key to solving integer problems accurately.