Subtracting integers

Adding integers we've done. Subtracting integers seems harder , what does $5 - (-3)$ even mean? , but it turns out to be the same operation, just written differently. The key trick: subtracting is the same as adding the opposite.

Concept

The rule

For any integers $a$ and $b$:

$a - b = a + (-b)$

In words: subtracting $b$ is the same as adding the opposite of $b$.

• If $b$ is positive, say $b = 5$, then $-b = -5$. So $a - 5 = a + (-5)$.
• If $b$ is negative, say $b = -3$, then $-b = +3$. So $a - (-3) = a + (+3) = a + 3$.

The strange-looking second case , "subtracting a negative is adding a positive" , is just a consequence of the rule. Picture it: if you remove a debt of ₹$3$ from your bank, your balance increases by $3$.

Why this rule works

Subtraction asks "how much do I add to $b$ to get $a$?" If I add $-b$ to both, I get $a - b$ on the left and $0$ on the right + $a - b$. Cleaner explanation: subtracting $b$ undoes the previous addition of $b$. If $+b$ moves you right, then $-b$ moves you left , that is, adds the opposite.

Worked-out rules of subtraction

Using the trick "subtraction = addition of opposite", you handle every case:

• $(+a) - (+b) = (+a) + (-b)$. Adding a positive to a negative , different-signs rule.
• $(+a) - (-b) = (+a) + (+b)$. Same-sign rule (both positive).
• $(-a) - (+b) = (-a) + (-b)$. Same-sign rule (both negative).
• $(-a) - (-b) = (-a) + (+b)$. Different-signs rule.

That's it , four cases, all reduced to addition.

Picture on a number line

To compute $a - b$, just move from $a$ in the opposite direction of $b$:

• Subtracting $+3$: move $3$ to the left.
• Subtracting $-3$: move $3$ to the right.

So $5 - (-3) = 5 + 3 = 8$. Start at $5$, move $3$ to the right, land at $8$.

Useful facts

• $a - 0 = a$. (Subtracting zero changes nothing.)
• $a - a = 0$. (A number minus itself is zero.)
• $0 - a = -a$. (Zero minus a number is its opposite.)
• $a - b$ is not generally equal to $b - a$. ($5 - 3 = 2$ but $3 - 5 = -2$.) Subtraction is not commutative.

Worked examples

Example 1. $7 - 3$.

• $7 + (-3) = 4$. (Same as basic subtraction.)

Example 2. $4 - 9$.

• $4 + (-9) = -5$.

Example 3. $-6 - 4$.

• $-6 + (-4) = -10$.

Example 4. $5 - (-3)$.

• $5 + (+3) = 8$.

Example 5. $-7 - (-2)$.

• $-7 + (+2) = -5$.

Example 6. $-10 - (-15)$.

• $-10 + (+15) = +5$.

Example 7. Word problem. The temperature was $-2°$C and dropped by $5°$. What is the new temperature?

• $-2 - 5 = -7$. So $-7°$C.

Example 8. The Mariana Trench is at $-11{,}000$ m and Mount Everest is at $+8{,}849$ m. How much higher is Everest than the trench bottom?

• $8849 - (-11000) = 8849 + 11000 = 19{,}849$ m.

Try it yourself

1. $9 - 4$.
2. $3 - 7$.
3. $-5 - 2$.
4. $6 - (-3)$.
5. $-8 - (-3)$.
6. $-12 - (-15)$.
7. The temperature drops by $7°$ from $4°$C. What is the new temperature?
8. Bela is on floor $+3$ and rides down $8$ floors. What floor?
9. Brijesh has a debt of $200$. He pays back $150$. What is his balance?
10. Tricky: find $x$ if $x - (-7) = 2$.

Activity

Two-step temperatures. Pick any city in India. Look up its temperatures for $5$ days last winter (you can use approximate values: Delhi $4°$C, Shimla $-2°$C, Srinagar $-5°$C, etc.). For each pair of days, calculate the difference in temperature. Express it as a signed integer. Which pair has the biggest difference?