31.04 - The product

The product of two numbers

Multiplying two numbers together is called the product.

The product is usually mathematically denoted as an x symbol: $\times$

The two numbers used in a product are called the factors.

The left number of the product will be added together the amount of times set by the right number (or the left number times the right number).

For example:

• The product of 2 and 6 is: $2\times 6$ or $2+2+2+2+2+2$
  • 2 and 6 are the factors
• The product of 1 and 3 is: $1\times 3$ or $1+1+1$
  • 1 and 3 are the factors
• The product of 3 and 4 is: $2\times 6$ or $3+3+3+3$
  • 3 and 4 are the factors

Products and negatives

As explained at The product of two numbers, multiplication is just how many times you add a certain number together. ($5\times 3=3+3+3+3+3=15$).

The same is true for negative factors: ($5\times -3=-3+-3+-3+-3+-3=-15$)
Writing this down for two negative factors ($-5$ and $-3$) is a bit more complex, but a way to do it is like:

$$\begin{array}{r}-5\times 2=-10\\ -5\times 1=-5\\ -5\times 0=0\\ -5\times -1=5\\ -5\times -2=10\\ -5\times -3=15\end{array}$$

Starting with a positive numbers and looking at the pattern moving down is one way. But there’s an easier way too. It is kind of like the method in The sum, difference, and negatives.

1. Take both factors in the equation: $-5$ and $-3$.
   1. If either number is positive, write down a $+$ in front of it: $5$ → $+5$
2. remove the numbers: $-$ and $-$
3. • $++$ will be yes yes, which is the same as saying yes ($+$)
   • $+-$ will be yes not, which is the same as saying not ($-$)
   • $-+$ will be not yes, which is the same as saying not ($-$)
   • $--$ will be not not, which is the same as saying yes ($+$)
4. The outcome of step 3 will decide whether the result is positive or negative: $-5\times -3=15$

Keep in mind that the second operator isnt actually an operator

It is an indicator that the number is negative. The dual operator trick i showed is just to make adding/subtracting with negative numbers easier.

Tip: If you have a hard time knowing the outcome at all

Since whether the outcome is positive or negative has been solved, you can remove all minuses, calculate the outcome, and add the minuses back.
$-5\times 3$ → $5\times 3=15$ → $-5\times 3=-15$

in case the equation has more than 2 numbers

If for example you have $5\times -3\times -4$, you apply the order of operations and do the equation in groups of 2 (in this case left to right).
OR, you could count the number of minuses and, if it’s an even amount, the solution is positive and vice versa.

next chapter

The next chapter will go over the quotient and fractions