Operations and Properties of Integers
Subjective Questions with Explanations
1. What are integers? Give examples.
Integers are whole numbers that include positive numbers, negative numbers, and zero, but exclude fractions and decimals. \[ \{-5, -2, -1, 0, 3, 7\} \] They are used to represent quantities such as debts, temperatures below zero, and gains.
2. Explain why the sum of two negative integers is always negative.
When two negative integers are added, their absolute values are added together and the negative sign is retained. \[ (-4) + (-6) = -(4 + 6) = -10 \] The result remains negative because both values move in the same negative direction.
3. Evaluate: \( -8 + 12 \) and explain your steps.
\[ -8 + 12 = 4 \] We subtract the smaller absolute value from the larger: \[ 12 - 8 = 4 \] Since the positive number has the larger absolute value, the answer is positive.
4. State and explain the commutative property of addition of integers.
The commutative property states that changing the order of addition does not change the result. \[ a + b = b + a \] Example: \[ 3 + (-5) = -5 + 3 = -2 \]
5. What is the additive identity of integers? Give an example.
The additive identity is zero $(0)$. Adding zero to any integer does not change its value. \[ 7 + 0 = 7 \quad \text{and} \quad -9 + 0 = -9 \]
6. Explain why the product of two negative integers is positive.
When two negative integers are multiplied, the negative signs cancel each other, resulting in a positive value. \[ (-4) \times (-3) = 12 \] This rule keeps multiplication patterns consistent.
7. Evaluate: \( (-3) \times 5 - (-10) \).
First multiply: \[ (-3) \times 5 = -15 \] Then subtract: \[ -15 - (-10) = -15 + 10 = -5 \]
8. State the distributive property and illustrate with an example.
The distributive property states: \[ a(b + c) = ab + ac \] Example: \[ -2(3 + 4) = -2 \times 7 = -14 \] \[ (-2 \times 3) + (-2 \times 4) = -6 - 8 = -14 \]
9. What is absolute value? Give examples.
Absolute value is the distance of a number from zero on the number line, without considering direction. \[ |-7| = 7 \quad \text{and} \quad |5| = 5 \] Absolute value is always non-negative.
10. Explain the closure property of integers under addition.
The closure property under addition states that the sum of any two integers is also an integer. \[ -6 + 4 = -2 \] Since $-2$ is an integer, integers are closed under addition.
