Integers
ICSE Class 7 · Mathematics · Chapter 1
What this chapter covers
- Integers and the number line
- Adding and subtracting integers
- Multiplication of integers
- Division of integers
- Properties of addition and subtraction
- Properties of multiplication and division
- Word problems on integers
Key points
- Integers are the numbers ldots, -3, -2, -1, 0, 1, 2, 3, ldots; zero is neither positive nor negative.
- On a number line, a number to the right is greater; of two negative integers, the one closer to zero is greater.
- Subtracting an integer is the same as adding its additive inverse.
- In multiplication and division, like signs give a positive answer and unlike signs give a negative answer.
- A product with an even number of negative factors is positive; with an odd number, it is negative.
- Any integer times 0 is 0; 0 divided by a non-zero integer is 0; division by 0 is not defined.
- Addition and multiplication of integers are commutative and associative; subtraction and division are not.
- 0 is the additive identity, 1 is the multiplicative identity, and every integer a has the additive inverse -a.
- Multiplication distributes over addition and subtraction: a × (b + c) = a × b + a × c.
- In word problems, choose a positive direction first, then write each quantity as an integer.
Questions answered
Find [(-3) × 4] × (-2) and (-3) × [4 × (-2)]. Which property of multiplication do the two answers show?
(-12) × (-2) = 24 and (-3) × (-8) = 24. The answers are equal, which shows that multiplication of integers is associative.
Which is greater, -17 or -71? Give a reason.
-17 is greater, because it is closer to zero and lies to the right of -71 on the number line.
Find (-28) + 13 - (-9).
(-28) + 13 + 9 = -6.
Find (a) (-13) × 4 (b) (-6) × (-15) (c) (-1) × (-1) × (-1) × (-1) × (-1).
(a) -52 (b) 90 (c) -1, since there are five negative factors.
Find (a) (-96) ÷ 8 (b) (-121) ÷ (-11) (c) 0 ÷ (-25).
(a) -12 (b) 11 (c) 0.
Why is (-9) ÷ 0 not defined?
No number multiplied by 0 gives -9, because every number multiplied by 0 gives 0.
Show with one example that subtraction of integers is not commutative.
(-2) - 6 = -8, but 6 - (-2) = 8; the results differ.
Name the additive inverse of -34 and the multiplicative identity. Does 5 have a multiplicative inverse among the integers?
The additive inverse of -34 is 34; the multiplicative identity is 1. No integer multiplied by 5 gives 1, so 5 has no multiplicative inverse among the integers; its inverse is the fraction (1)/(5).