Addition and subtraction represent the two simplest arithmetic operations.
\color{cyan}{\textnormal{Addition}} is the act of aggregating two numbers together.
Addition answers: “What happens when I combine one group of things with another group of things to make a larger group of things?”
For instance, if I have 5 apples in one basket, and 10 apples in another basket, and combine them into a bigger basket, how many apples would I have?
Well, the first step is to realize that the final basket would have the 5 apples from the first basket, and 10 apples from the second. You could perform this by taking 5 apples, dumping 10 more apples in the basket and counting each. You will get 15 apples, showing that addition is a simple expansion of counting.
Alternatively, you could perform this much easier by adding the two quantities together. That would result in 15 total apples!
The addition operation is represented by the + sign. To represent the situation above, you would write:
5 + 10
This is spoken as “Five plus ten.”
Which represents the actual act of adding the two numbers. The next step when writing an expression is to \color{cyan}{\textnormal{simplify}} it. That is, make the expression as consolidated as possible. In this case, you would add the two numbers to get 15. You can write this statement as the equation:
5 + 10 = 15
Or as a series of steps:
\begin{aligned} 5 + 10\\ = 15 \end{aligned}
This is spoken as “Five plus ten equals fifteen.”
Addition refers to their terms as \color{cyan}{\textnormal{addends}} and the result of the operation is called the \color{cyan}{\textnormal{sum}}. That is, for 2 + 3 = 5 the addends are 2 and 3, while 5 represents the sum. So, it can be modeled as:
\text{addend} + \text{addend} = \text{sum}
The operands is sometimes referred to as \color{cyan}{\textnormal{summands}}
Long addition is a method of performing addition by hand where the two addends are placed vertically. The addition is performed by “column” where numbers are aligned in a particular manner to enable adding by digit.
\begin{aligned} \phantom{+}100 \\ \underline{+\phantom{1}99} \\ \phantom{+}199 \\ \end{aligned}
Whenver a digit “overflows”, that is, when two digits added together exceeds the maximum value for a single digit (such as, 5+5 = 10; a single digit can only represent a maximum value of 9), the remaining value must be “carried over” to the next digit. Here’s an example of long addition where you would need to utilize such a carry-over technique.
\begin{aligned} \phantom{+}852 \\ \underline{+\phantom{0}299} \\ \phantom{+}1151 \\ \end{aligned}
To understand how this works step by step, I will reproduce the steps necessary as individual “snapshots” of this process, with comments between each to describe what was done immediately above it.
\begin{aligned} \phantom{00}_1\phantom{0}\\ \phantom{+}852 \\ \underline{+\phantom{0}299} \\ \phantom{+115}1 \\ \end{aligned}
\text{2 + 9 = 11, so you must carry the 1 to the next column. Notice the 1 at the top.} \\
\begin{aligned} \phantom{0}_1\phantom{00}\\ \phantom{+}852 \\ \underline{+\phantom{0}299} \\ \phantom{+11}51 \\ \end{aligned}
\text{1 + 5 + 9 = 15, so you must carry again!} \\
\begin{aligned} \phantom{}_1\phantom{000}\\ \phantom{+}852 \\ \underline{+\phantom{0}299} \\ \phantom{+1}151 \\ \end{aligned}
\text{1 + 8 + 2 = 11; another carry!} \\
\begin{aligned} \phantom{+}852 \\ \underline{+\phantom{0}299} \\ \phantom{+}1151 \\ \end{aligned}
\text{Simply bring the final carried 1 down. Congrats!} \\
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