Multiplication is repeated addition. It is an incredibly helpful operation that can be used to compute rates (like distance when travelling at a certain speed, or how much your wages would be after an 8-hour shift).
Multiplication is essentially repeated addition.
Multiplication answers: “What if I add a number to itself a certain number of times?”
This sounds strange, so let’s look at an example. Say you make $15 an hour, and work 4 hours. How much money did you make?
You could represent this as chunks of hours, added together, like this:
15 + 15 + 15 + 15 = 60
Imagine if you worked 8 hours, or wanted to summarize a whole month of work (say, 160 hours). Writing that whole situation out for 160 hours would be quite redundant. Multiplication makes this a lot simpler. To model the original situation of $15 an hour for 4 hours, you could instead write:
15 \times 4
Now multiplication actually has multiple symbols (go figure), which can be any of the following: \times, \cdot, *, etc. While they all look different, they represent the same operation.
Multiplication refers to their operands as \color{cyan}{\textnormal{factors}}. Sometimes, you will hear \color{cyan}{\textnormal{multiplicand}} and \color{cyan}{\textnormal{multiplier}}, which refer to the first and second terms respectively. The result of multiplication is known as the \color{cyan}{\textnormal{product}}. That is, multiplication can be modeled as either:
\text{factor} \cdot \text{factor} = \text{product} \text{multiplicand} \cdot \text{multiplier} = \text{product}
There are a few conventions that should be followed when writing these operations.
When multiplication is presented with parentheses, you can omit the symbol:
4 \cdot (3 + 2) = 4(3+2)
When multiplication is presented with letters or non-numerical symbols, you can omit the symbol:
4 \cdot x = 4x
Typically, to avoid confusion, if you are using an kind of letters or non-numerical symbols, you would want to avoid using the \times symbol, since (especially with hand-writing) that could be confused with the x.
\color{cyan}{\textnormal{Long Multiplication}} is a common way of performing multiplication by-hand. It is of a similar form as long subtraction and long addition. In fact, it will utilize long addition as a means of computing the final result.
Let’s start with a simple example:
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times}1508 \\ \end{aligned}
Now, similar to the previous chapters, I will walk through the problem in a series of snapshots to show how long multiplication works.
The general algorithm (process) to do this is to:
\begin{aligned} &\phantom{000}_1\phantom{0}\\ &\phantom{\times00}52 \\ &\underline{\times\phantom{00}29} \\ &\phantom{\times000}8 \\ &\end{aligned}
\text{Notice that you must carry, since }9 \cdot 2 = 18
\begin{aligned} \phantom{0}_4\phantom{00}\\ \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times00}68 \\ \end{aligned}
\text{You must consider the carry. We know that }9 \cdot 5 = 45 \text{, but then include the carry: }45+1 = 46.\\ \phantom{\n} \\ \text{Now, the ones place is done! Bring the carry down!}
\begin{aligned} \phantom{00}\phantom{00}\\ \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \end{aligned}
\text{Then, move onto the next digit.}
\begin{aligned} \phantom{00}\phantom{00}\\ \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \phantom{\times000}0 \\ \end{aligned}
\text{The first step is to add the 0 as a placeholder.}
\begin{aligned} \phantom{00}\phantom{00}\\ \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \phantom{\times00}40 \\ \end{aligned}
\text{There's no carry here, because }2 \cdot 2 = 4
\begin{aligned} \phantom{}_1\phantom{000}\\ \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \phantom{\times0}040 \\ \end{aligned}
\text{We need to carry the 1, since }2 \cdot 5 = 10
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \phantom{\times}1040 \\ \end{aligned}
\text{Once you bring the carry down, you can perform long addition!}
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \underline{+\phantom{0}1040} \\ \phantom{0000}8 \end{aligned}
\text{The addition is: } 8+0=8 \text{, which results in no carry. Continue:}
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{0}_1\phantom{00}\\ \phantom{\times0}468 \\ \underline{+\phantom{0}1040} \\ \phantom{000}08 \end{aligned}
\text{This step's addition is }6 + 4 = 10.\text{ Carry the 1 to the next digit, and continue.}
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{0}_1\phantom{00}\\ \phantom{\times0}468 \\ \underline{+\phantom{0}1040} \\ \phantom{00}508 \end{aligned}
\text{This step's addition must include the carry: }4 + 1 + 0 = 5
\begin{aligned} \phantom{\times0}52 \\ \underline{\times\phantom{00}29} \\ \phantom{\times0}468 \\ \underline{+\phantom{0}1040} \\ \phantom{0}1508 \end{aligned}
\text{Congratulations! You're finished! We computed that:}
\boxed{52 \cdot 29 = 1508}
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