1.01a - Adding and subtracting integers

1.01a - Adding and subtracting integers

Can subtracting a number make your answer bigger? Learn to add and subtract positive and negative integers, using a number line to explain every sign.

Numbers on both sides of zero

Can −4−(−7)-4-(-7) be larger than −4-4? Keep that question in mind: the two minus signs do different jobs, and understanding those jobs will unlock the calculation.

Start with something easier to picture. A temperature of 4 °C4\,\mathrm{°C} is four degrees above zero; −4 °C-4\,\mathrm{°C} is four degrees below zero. The number without its sign tells you the distance from zero. The sign tells you which side of zero to use.

[DIAGRAM: asset_name: Signed integers on a number line; asset_slug: signed-number-line; caption: Numbers increase to the right; opposite integers are equally far from zero.; description: A horizontal number line from negative eight to positive eight. Negative four and positive four are marked, with equal four-unit distances from zero. A right-pointing arrow shows the direction of increasing value.]
Numbers increase to the right

Integer

An integer is a whole number, its negative, or zero. For example, −12-12, 00 and 88 are integers. 2.52.5 and 12\frac12 are not integers.

On a number line, moving right increases the value; moving left decreases it. So −2-2 is greater than −7-7: it is further right, even though its distance from zero is smaller. Zero is neither positive nor negative.

Two numbers the same distance from zero on opposite sides are opposites. The opposite of 66 is −6-6, and the opposite of −6-6 is 66. Zero is its own opposite. You can write a positive integer as +6+6, but usually we leave off the plus sign and write 66.

In 8−38-3, the minus sign is an instruction to subtract. In −3-3, it belongs to the number: negative three. Brackets make that distinction visible in a calculation such as 8+(−3)8+(-3).

Key idea

A minus sign can name a negative number or tell you to subtract. Read its job before deciding which way to move.

Use the number line to check this claim before choosing an answer.

Adding signed integers

Adding tells you to combine the starting value with a change: a positive change moves right, and a negative change moves left.

For example, starting at −3-3 and adding 55 means moving five units right: three units reach zero and the remaining two reach 22. Starting at −3-3 and adding −5-5 means moving five units left, to −8-8. The second number decides the direction; the starting number does not.

Sum

The sum of two numbers is the result of adding them. Adding a positive integer increases the starting value; adding a negative integer decreases it. Adding zero leaves it unchanged.

You do not have to draw every small step. When a move crosses zero, split it into the distance to zero and the distance left to travel.

Worked example 1: crossing zero

Question. Work out −5+8-5+8.

Plan. Start at −5-5, move right, and split the eight-unit move at zero.

−5+8=−5+5+3split 8 into 5 and 3=0+3five units reach zero=3three units remain\begin{aligned} -5+8 &= -5+5+3 && \text{split 8 into 5 and 3} \\ &= 0+3 && \text{five units reach zero} \\ &= 3 && \text{three units remain} \end{aligned}

Answer. 33.

Check. The positive amount is larger. The difference between the distances from zero is 8−5=38-5=3, so the answer should be positive three.

Now change only the sign of the number being added. The starting point and size of the move stay the same; the direction reverses.

Worked example 2: moving further below zero

Question. Work out −5+(−8)-5+(-8).

Plan. Start five units left of zero and move another eight units left.

5+8=13total distance left of zero−5+(−8)=−13the position is negative\begin{aligned} 5+8 &= 13 && \text{total distance left of zero} \\ -5+(-8) &= -13 && \text{the position is negative} \end{aligned}

Answer. −13-13.

Check. Moving eight units right from −13-13 returns to −5-5. That reverses the change we made.

The signs explain the answer more reliably than a slogan about “two minuses”.

Watch out

The line −5+(−8)=13-5+(-8)=13 is wrong. Both numbers are negative, so their sum lies further left of zero: −5+(−8)=−13-5+(-8)=-13. Two negative numbers do not make a positive sum.

Try the same move with a different negative amount.

Subtracting a positive integer

Subtracting a positive integer moves you left, and the number line continues past zero.

You already know 9−4=59-4=5. The same four-unit move works from any starting point: 2−4=−22-4=-2, because two units reach zero and two more reach −2-2. There is no stopping rule at zero.

The difference is the result of subtraction. The order matters: 2−42-4 and 4−24-2 have different answers. If a question asks how far apart two values are, the distance is non-negative; first decide which subtraction represents that distance.

Worked example 3: subtracting past zero

Question. Work out 7−97-9.

Plan. Move nine units left from 77, using zero as a stopping point on the way.

7−9=7−7−2split 9 into 7 and 2=0−2seven units reach zero=−2two more units left\begin{aligned} 7-9 &= 7-7-2 && \text{split 9 into 7 and 2} \\ &= 0-2 && \text{seven units reach zero} \\ &= -2 && \text{two more units left} \end{aligned}

Answer. −2-2.

Check. Adding back the amount subtracted gives −2+9=7-2+9=7, the original value.

Change only the sign of the starting number. Subtracting nine still moves left.

Worked example 4: starting below zero

Question. Work out −7−9-7-9.

Plan. Start seven units left of zero and move another nine units left.

7+9=16total distance left of zero−7−9=−16the position is negative\begin{aligned} 7+9 &= 16 && \text{total distance left of zero} \\ -7-9 &= -16 && \text{the position is negative} \end{aligned}

Answer. −16-16.

Check. −16+9=−7-16+9=-7. Adding back nine restores the starting value.

An answer can be negative even when the number being subtracted is positive.

Watch out

The line 7−9=27-9=2 gives the distance between the numbers, but not this subtraction. From 77, moving nine units left ends at −2-2, so 7−9=−27-9=-2.

Reverse your move to check the next calculation.

Subtracting a negative integer

Subtracting reverses addition, so subtracting a negative change reverses a move to the left.

Adding −7-7 moves seven units left. Undoing that addition must move seven units right. That is why subtracting −7-7 has the same effect as adding 77. This is about the operation and the sign together, not about any two minus signs appearing somewhere in a calculation.

[DIAGRAM: asset_name: Adding and subtracting a negative number; asset_slug: subtracting-negative; caption: Adding negative seven moves left; subtracting negative seven reverses that move.; description: Two aligned horizontal number lines. On the upper line an arrow goes from negative four to negative eleven, labelled add negative seven. On the lower line an arrow goes from negative four to positive three, labelled subtract negative seven. Each arrow has length seven units.]
Adding negative seven moves left

Key idea

To subtract an integer, add its opposite. For example, 6−(−2)=6+26-(-2)=6+2, while 6−2=6+(−2)6-2=6+(-2). Subtracting zero leaves the number unchanged.

We can now answer the opening question by reading both minus signs carefully.

Worked example 5: subtraction makes the answer bigger

Question. Work out −4−(−7)-4-(-7).

Plan. Replace subtraction of negative seven with addition of its opposite.

−4−(−7)=−4+7add the opposite of −7=−4+4+3split the move at zero=3three units beyond zero\begin{aligned} -4-(-7) &= -4+7 && \text{add the opposite of }-7 \\ &= -4+4+3 && \text{split the move at zero} \\ &= 3 && \text{three units beyond zero} \end{aligned}

Answer. 33.

Check. Add back the number subtracted: 3+(−7)=−43+(-7)=-4. We recover the starting value, so the subtraction is consistent.

Subtracting a negative always increases the starting value, but it need not make the answer positive. Change only the size of the number subtracted to see why.

Worked example 6: a larger answer can still be negative

Question. Work out −4−(−2)-4-(-2).

Plan. Add the opposite of negative two, then move right from −4-4.

−4−(−2)=−4+2add the opposite of −2=−2two units right\begin{aligned} -4-(-2) &= -4+2 && \text{add the opposite of }-2 \\ &= -2 && \text{two units right} \end{aligned}

Answer. −2-2.

Check. −2+(−2)=−4-2+(-2)=-4. Adding back the number subtracted restores the original value. Also, −2-2 is greater than −4-4, although both are negative.

The surprise is now explained: subtraction can make a number larger, because the number subtracted can be negative. Later, this same idea will help you rearrange equations without losing a sign.

Watch out

Do not replace −4−(−2)-4-(-2) with −4−2-4-2. That changes the number being subtracted from negative two to positive two. Keep the brackets until you have written the equivalent addition: −4+2-4+2.

Use the inverse check as well as the sign rule.

Quick Recap

  • Integers include negative whole numbers, zero and positive whole numbers. Zero is neither positive nor negative.
  • Adding a positive integer moves right; adding a negative integer moves left.
  • To subtract an integer, add its opposite. Subtraction order matters.
  • Check a subtraction by adding back the number you subtracted.

Yes, subtraction can make an answer bigger: −4−(−7)=3-4-(-7)=3, because subtracting negative seven moves seven units right.