Operations with Integers

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CLASS VII Mathematics ~6 marks/year Ch 10 of 15
Operations with Integers

Class 7 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • Integers are the whole numbers together with their negatives: $\dots,-3,-2,-1,0,1,2,3,\dots$ — written as $\mathbb{Z}$.
  • On the number line, numbers increase to the right and decrease to the left; every positive has a mirror-image negative.
  • Addition & subtraction: same signs add and keep the sign; different signs subtract and keep the bigger number's sign. Subtracting is adding the opposite.
  • Multiplication & division of signs: like signs give a $+$ result, unlike signs give a $-$ result.
  • Integers are closed under $+$, $-$, $\times$ (but not $\div$) and obey commutative, associative and distributive laws — with special roles for $0$ and $1$.
  • Integers model real life: temperatures, depths, money owed, floors below ground, and scores above/below par.
  • Weightage: ~6 marks/year — sign rules, word problems, and using properties to simplify.
Detailed Notes

1. What integers are

The counting numbers $1,2,3,\dots$ are natural numbers. Adding $0$ gives whole numbers $0,1,2,3,\dots$ But many everyday quantities go below zero — a temperature of $5$ degrees below freezing, a debt of $200$ rupees, a basement floor. To describe these we need negative numbers.

Integers are all the whole numbers together with their negatives:

$\mathbb{Z}=\{\dots,-3,-2,-1,0,1,2,3,\dots\}$

Numbers like $1,2,3,\dots$ are positive integers; $-1,-2,-3,\dots$ are negative integers. The number $\mathbf{0}$ is an integer that is neither positive nor negative. Two integers like $5$ and $-5$ that are the same distance from $0$ on opposite sides are called opposites (or additive inverses) of each other.

2. Integers on the number line

Draw a line, mark $0$ in the middle, place $1,2,3,\dots$ to the right and $-1,-2,-3,\dots$ to the left. This number line is the key to comparing integers and to seeing what addition and subtraction do.

On the number line, a number on the right is always greater; a number on the left is always smaller. So $-1>-5$, and every positive integer is greater than every negative integer.

Examples of ordering: $-7<-2<0<3<8$. Be careful with negatives — $-2$ is greater than $-9$ because it is closer to $0$ (further to the right).

3. Adding integers

Think of moving on the number line: adding a positive number moves right; adding a negative number moves left.

The shortcut rules:

  • Same signs: add the numbers and keep the common sign. $(+6)+(+4)=+10$; $(-6)+(-4)=-10$.
  • Different signs: subtract the smaller value from the larger value, and keep the sign of the number with the larger value. $(+9)+(-5)=+4$; $(-9)+(+5)=-4$.
A number plus its opposite is always zero: $a+(-a)=0$. For example $7+(-7)=0$.
Worked example — add $(-8)+(+3)$

Different signs. Larger value is $8$, smaller is $3$: $8-3=5$. The larger value $8$ is negative, so the answer keeps a minus sign: $(-8)+(+3)=-5$.

Worked example — add $(-15)+(-7)$

Same sign (both negative). Add: $15+7=22$. Keep the common sign: $(-15)+(-7)=-22$.

4. Subtracting integers

Subtraction is turned into addition: to subtract an integer, add its opposite.

$a-b=a+(-b)$   — change the sign of the number being subtracted, then add.
Worked example — compute $5-(-3)$

Subtracting $-3$ means adding its opposite $+3$: $5-(-3)=5+3=8$. Subtracting a negative makes the result bigger.

Worked example — compute $(-7)-(+4)$

Add the opposite of $+4$, which is $-4$: $(-7)-(+4)=(-7)+(-4)=-11$.

Worked example — temperature drop

At night the temperature was $3°$C; it fell by $8°$. New temperature $=3-8=3+(-8)=-5°$C, i.e. $5$ degrees below zero.

5. Multiplying integers — the sign rules

Multiplication of integers follows two simple sign rules, then you just multiply the values.

$(+)\times(+)=(+)$  ·  $(-)\times(-)=(+)$
$(+)\times(-)=(-)$  ·  $(-)\times(+)=(-)$

In words: like signs give plus, unlike signs give minus. First decide the sign, then multiply the numbers as usual.

Worked example — multiply several integers

$(-4)\times(+6)=-24$ (unlike signs). $(-5)\times(-7)=+35$ (like signs). $(+3)\times(+9)=+27$ (like signs).

Counting the minus signs: when several integers are multiplied, count how many factors are negative.

An even number of negative factors gives a positive product; an odd number of negative factors gives a negative product.
Worked example — $(-2)\times(-3)\times(-4)$

There are three negative factors (odd), so the product is negative. Values: $2\times3\times4=24$. Hence $(-2)\times(-3)\times(-4)=-24$.

6. Dividing integers

Division uses exactly the same sign rules as multiplication.

$(+)\div(+)=(+)$  ·  $(-)\div(-)=(+)$
$(+)\div(-)=(-)$  ·  $(-)\div(+)=(-)$
Worked example — divide integers

$(-36)\div(+9)=-4$ (unlike signs). $(-48)\div(-6)=+8$ (like signs). $(+45)\div(-5)=-9$ (unlike signs).

Special facts: $\dfrac{0}{a}=0$ for any non-zero $a$, but division by zero is not defined — $\dfrac{a}{0}$ has no meaning. Also $a\div 1=a$ and $a\div a=1$ (for $a\neq0$).

7. Properties of integers

Integers obey the same friendly rules as whole numbers — and a few that whole numbers cannot.

Closure

Add, subtract or multiply any two integers and you always get an integer. So integers are closed under $+$, $-$ and $\times$. They are not closed under division: $(-7)\div(2)$ is not an integer.

Commutative property

Order does not matter for addition and multiplication: $a+b=b+a$ and $a\times b=b\times a$. But subtraction and division are not commutative: $5-3\neq3-5$.

Associative property

Grouping does not matter for addition and multiplication: $(a+b)+c=a+(b+c)$ and $(a\times b)\times c=a\times(b\times c)$.

Distributive property

$a\times(b+c)=a\times b+a\times c$   and   $a\times(b-c)=a\times b-a\times c$

Identity elements

0 is the additive identity: $a+0=a$. 1 is the multiplicative identity: $a\times1=a$. Also, multiplying by zero always gives zero: $a\times0=0$.

Worked example — using distributivity to compute fast

$(-25)\times(102)=(-25)\times(100+2)=(-25)\times100+(-25)\times2=-2500+(-50)=-2550.$ The distributive law turns a hard product into two easy ones.

8. Order of operations (BODMAS) with integers

When an expression mixes operations, work in this order: Brackets, then Of/orders, then Division and Multiplication (left to right), then Addition and Subtraction (left to right). The sign rules apply throughout.

Worked example — simplify $(-6)+4\times(-3)-(-8)$

Multiply first: $4\times(-3)=-12$. Now $(-6)+(-12)-(-8)=(-18)+8=-10$.

Worked example — simplify $15-(8-12)\div(-2)$

Bracket first: $8-12=-4$. Then divide: $(-4)\div(-2)=2$. Finally $15-2=13$.

9. Integers in real life — word problems

Integers describe quantities with direction: above/below, gain/loss, forward/backward.

Worked example — bank account

Priya has ₹500. She spends ₹800. Write the spending as $-800$. Balance $=500+(-800)=-300$, i.e. she is ₹300 in debt (overdrawn).

Worked example — a quiz with negative marking

Each correct answer scores $+5$ and each wrong one scores $-2$. A student gets $8$ correct and $5$ wrong. Score $=8\times(+5)+5\times(-2)=40+(-10)=30$ marks.

Worked example — a diver and a drone

A diver is at $-30$ m (below sea level) and a drone is at $+45$ m (above). The vertical distance between them $=45-(-30)=45+30=75$ m.

10. Common mistakes to avoid

  • Treating "subtracting a negative" as subtraction — remember $5-(-3)=5+3=8$.
  • Forgetting the sign rule for products: unlike signs give a minus, not a plus.
  • Saying $-2<-9$ — it is the other way round: $-2>-9$ because $-2$ is closer to $0$.
  • Writing $a\div0=0$ — division by zero is undefined, not zero.
  • Adding before multiplying — always follow BODMAS.
  • Assuming subtraction is commutative — $a-b\neq b-a$ in general.

11. Quick revision checklist

  • Integers: $\dots,-2,-1,0,1,2,\dots$; right means greater on the number line.
  • Addition: same sign add & keep sign; different sign subtract & keep bigger one's sign.
  • $a-b=a+(-b)$ — subtraction = add the opposite.
  • Sign rule: like signs $\to+$, unlike signs $\to-$ (for $\times$ and $\div$).
  • Even number of negatives $\to+$ product; odd number $\to-$ product.
  • Properties: closure ($+,-,\times$), commutative & associative ($+,\times$), distributive; $0$ and $1$ identities; never divide by $0$.
Practice MCQs
1. The value of $(-8)+(+3)$ is:
  1. $11$
  2. $-11$
  3. $5$
  4. $-5$
Answer: (D) Different signs: $8-3=5$, keep the sign of the larger value ($-8$), so $-5$.
2. $5-(-3)$ equals:
  1. $2$
  2. $8$
  3. $-8$
  4. $-2$
Answer: (B) Subtracting $-3$ adds $+3$: $5+3=8$.
3. The product $(-6)\times(-7)$ is:
  1. $42$
  2. $-42$
  3. $13$
  4. $-13$
Answer: (A) Like signs give a positive product: $+42$.
4. $(-2)\times(-3)\times(-1)$ equals:
  1. $6$
  2. $-6$
  3. $5$
  4. $-5$
Answer: (B) Three (odd) negative factors give a negative product: $-6$.
5. $(-45)\div(+9)$ equals:
  1. $5$
  2. $-5$
  3. $-36$
  4. $36$
Answer: (B) Unlike signs give a negative result: $-5$.
6. Which integer is greatest?
  1. $-9$
  2. $-2$
  3. $-15$
  4. $-7$
Answer: (B) $-2$ is closest to $0$, hence the greatest.
7. The additive inverse of $-12$ is:
  1. $0$
  2. $12$
  3. $-12$
  4. $1$
Answer: (B) Its opposite is $12$, since $-12+12=0$.
8. Integers are NOT closed under:
  1. Addition
  2. Subtraction
  3. Multiplication
  4. Division
Answer: (D) Dividing two integers may give a non-integer, e.g. $7\div2$.
9. $(-25)\times(102)$ using distributivity equals:
  1. $-2550$
  2. $2550$
  3. $-2500$
  4. $-50$
Answer: (A) $(-25)(100+2)=-2500+(-50)=-2550$.
10. Simplify $(-6)+4\times(-3)$:
  1. $-6$
  2. $-18$
  3. $6$
  4. $18$
Answer: (B) Multiply first: $4\times(-3)=-12$; then $(-6)+(-12)=-18$.
11. The value of $a\times 0$ for any integer $a$ is:
  1. $a$
  2. $1$
  3. $0$
  4. undefined
Answer: (C) Any integer multiplied by $0$ is $0$.
12. $(-15)+(-7)$ equals:
  1. $-8$
  2. $8$
  3. $22$
  4. $-22$
Answer: (D) Same sign: add values $15+7=22$, keep the minus: $-22$.
13. The multiplicative identity for integers is:
  1. $0$
  2. $1$
  3. $-1$
  4. $10$
Answer: (B) $a\times1=a$, so $1$ is the multiplicative identity.
14. A submarine is at $-250$ m. It rises by $90$ m. Its new position is:
  1. $-340$ m
  2. $-160$ m
  3. $160$ m
  4. $340$ m
Answer: (B) $-250+90=-160$ m.
15. Which statement is TRUE?
  1. $a-b=b-a$
  2. $a\div b=b\div a$
  3. $a+b=b+a$
  4. $a\div 0=0$
Answer: (C) Addition is commutative; subtraction and division are not, and dividing by $0$ is undefined.
Important Questions
Q1. State the sign rules for multiplication of integers and use them to find $(-9)\times(+4)$ and $(-9)\times(-4)$. (2 marks)
Answer: Like signs give a positive product, unlike signs give a negative product. So $(-9)\times(+4)=-36$ (unlike signs) and $(-9)\times(-4)=+36$ (like signs).
Q2. The temperature at $6$ a.m. was $-4°$C. By noon it rose by $11°$C, and by night it fell by $6°$C. Find the night temperature. (2 marks)
Answer: Start $-4°$C. After rise: $-4+11=7°$C. After fall: $7-6=1°$C. The night temperature is $1°$C.
Q3. Verify the distributive property $a\times(b+c)=a\times b+a\times c$ for $a=-3,\ b=5,\ c=-2$. (3 marks)
Answer: LHS $=-3\times(5+(-2))=-3\times3=-9$. RHS $=(-3)\times5+(-3)\times(-2)=-15+6=-9$. LHS $=$ RHS, so the property holds.
Q4. In a test, $+4$ marks are given for each correct answer and $-1$ for each wrong answer. A student attempts all questions, gets $15$ correct and $5$ wrong. Find the total score. (3 marks)
Answer: Correct: $15\times(+4)=60$. Wrong: $5\times(-1)=-5$. Total $=60+(-5)=55$ marks.
Q5. Simplify $36\div(-9)+(-4)\times(3-7)$ using the correct order of operations. (3 marks)
Answer: Bracket: $3-7=-4$. Then $36\div(-9)=-4$ and $(-4)\times(-4)=16$. Finally $-4+16=12$.
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