Integers — NCERT Class 7: 31 flashcards
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1. What is the set of integers?
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$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$ — whole numbers with their negatives.
2. Is zero positive, negative, or neither?
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Zero is an integer but is neither positive nor negative.
3. On a number line, which direction means increasing value?
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Moving right increases value; moving left decreases it.
4. Which is smaller: $-9$ or $-4$?
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$-9 < -4$ because $-9$ lies further left on the number line.
5. Define absolute value $|x|$.
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$|x|$ is the distance of $x$ from $0$ on the number line; it is always non-negative.
6. What is $|-6|$?
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$6$
7. Rule for adding two integers with the same sign?
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Add their absolute values and keep the common sign. Example: $(-3) + (-5) = -8$.
8. Rule for adding two integers with opposite signs?
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Subtract the smaller absolute value from the larger, and keep the sign of the integer with the larger absolute value. Example: $(-7) + 4 = -3$.
9. Rewrite subtraction as addition: $a - b = ?$
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$a - b = a + (-b)$
10. Compute $6 - 9$ using the addition rule.
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$6 + (-9) = -3$
11. Sign of the product of two integers with the same sign?
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Positive. $(+) \times (+) = (+)$ and $(-) \times (-) = (+)$.
12. Sign of the product of two integers with different signs?
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Negative. $(+) \times (-) = (-)$ and $(-) \times (+) = (-)$.
13. Compute $(-20) \div (-4)$.
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$5$ (same signs → positive quotient)
14. What is the additive identity for integers?
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$0$, because $a + 0 = a$ for every integer $a$.
15. What is the multiplicative identity for integers?
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$1$, because $a \times 1 = a$ for every integer $a$.
16. What is the additive inverse of an integer $a$?
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$-a$, because $a + (-a) = 0$.
17. If $a = -5$, what is $-a$?
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$-a = 5$, which is positive. The minus sign flips the sign of whatever is inside.
18. Are integers closed under division? Give a counterexample.
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No. $3 \div 4$ is not an integer, so integers are not closed under division.
19. State the distributive property for integers.
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$a \times (b + c) = a \times b + a \times c$
20. Give two real-life situations modelled by negative integers.
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Temperatures below zero, debts/bank debits, depths below sea level, floors below ground.
21. Why is $-3 \times (-4) = +12$?
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Multiplying by a negative reverses direction; the pattern $-3 \times 3 = -9$, $-3 \times 2 = -6$, $-3 \times 1 = -3$, $-3 \times 0 = 0$ forces $-3 \times (-1) = +3$, and so on.
22. What does $a \times 0$ equal for any integer $a$?
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$0$
23. Is $0 \div 5$ allowed? What is it?
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Yes, $0 \div 5 = 0$. (But division *by* zero is not allowed.)
24. Is $5 \div 0$ allowed?
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No. Division by zero is not defined.
25. Which is greater: $-2$ or $-7$?
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$-2 > -7$ because $-2$ is to the right of $-7$ on the number line.
26. State the commutative property of addition for integers.
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$a + b = b + a$
27. State the associative property of addition for integers.
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$(a + b) + c = a + (b + c)$
28. Compute $5 + (-8)$ using the number line.
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Start at $5$, move $8$ steps left → $-3$.
29. Compute $(-4) - (-7)$.
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$(-4) + 7 = 3$
30. Are integers closed under addition, subtraction and multiplication?
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Yes. The sum, difference and product of any two integers is always an integer.
31. What is the sign of $(-3) \times 4 \times (-2)$?
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Positive, because there are two negative factors (an even number).
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