Multiplying Integers on a Number Line

A positive first factor tells you how many groups to add. So $3 \times (-2)$ means $3$ groups of $-2$: start at $0$ and add $-2$ three times.

Adding $-2$ moves you $\lvert -2 \rvert = 2$ units in the direction of $-2$, which is left. (The absolute value $\lvert -2 \rvert$ is the distance of $-2$ from $0$.)

• Start at $0$.
• Add the first group: $0 + (-2) = -2$.
• Add the second group: $-2 + (-2) = -4$.
• Add the third group: $-4 + (-2) = -6$.

So $3 \times (-2) = -6$. The number line below marks these four positions: the start $0$, the in-between positions $-2$ and $-4$, and the result $-6$. It marks positions only. The direction (left) and the distance ($2$ units for each group) come from the steps above.

What can a negative first factor mean? You cannot add a group negative three times, so ask what $-3$ groups must do.

Groups of the same number combine when you add their counts: $3$ groups of $2$ plus $4$ groups of $2$ make $7$ groups of $2$. This is the distributive property, and multiplication of integers is defined so that it stays true when a count is negative. So $3$ groups of $2$ plus $-3$ groups of $2$ make $3 + (-3) = 0$ groups:

$3 \times 2 + (-3) \times 2 = (3 + (-3)) \times 2 = 0 \times 2 = 0$

Two numbers that sum to $0$ are called opposites, such as $6$ and $-6$. So $(-3) \times 2$ must be the opposite of $3 \times 2 = 6$, which is $-6$.

Subtracting groups gives exactly that opposite. When you subtract groups from $0$, you land on the opposite of where adding the same groups would take you, whatever the group is. So a negative first factor tells you to subtract groups from $0$, and its absolute value tells you how many.

For $(-3) \times 2$, subtract $2$ three times. Subtracting $2$ adds its additive inverse (its opposite), $-2$, so each subtraction moves you $2$ units left, the direction of $-2$.

• Start at $0$.
• Subtract the first group: $0 - 2 = -2$.
• Subtract the second group: $-2 - 2 = -4$.
• Subtract the third group: $-4 - 2 = -6$.

So $(-3) \times 2 = -6$. You pass through the same positions that are marked on the number line for $3 \times (-2)$: $0$, $-2$, $-4$ and $-6$.

Compare the two number lines. The first marks $0$, $-2$, $-4$ and $-6$. The second marks $0$, $2$, $4$ and $6$. Adding $3$ groups of $-2$ took you left to $-6$, and subtracting $3$ groups of $-2$ took you right to $6$. Both times you made $3$ moves of $2$ units, so both results are $3 \times 2 = 6$ units from $0$. Only the direction changed, because subtracting a group adds its additive inverse, which points the opposite way.

This reasoning works for any two nonzero integer factors: the absolute value of the first factor is the number of moves, and the absolute value of the second factor is the length of each move. The table shows the direction in each case, using the sizes $3$ and $2$, and then what happens when a factor is $0$.

Takeaway: multiplying two integers means making equal moves that start at $0$ on the number line, and the product is where you end. A positive first factor adds groups. A negative first factor subtracts groups, and subtracting a group adds its additive inverse, so the move goes the opposite way. That is why, when no factor is $0$, you can multiply the absolute values and then make the product negative exactly when the number of negative factors is odd. When any factor is $0$, the product is $0$. Next, you will use these ideas to divide integers.