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Lesson 1: Intro to adding negative numbers

# Negative numbers: addition and subtraction FAQ

## When and where did people start using negative numbers?

There is no definitive answer to when and where people started using negative numbers, as different cultures may have developed or encountered them independently, and the historical evidence is often fragmentary, ambiguous, or contested. However, here are some possible milestones in the history of negative numbers.
The earliest known use of negative numbers in a mathematical context is probably in the Chinese text Jiuzhang Suanshu (Nine Chapters on the Mathematical Art), which dates from the ${1}^{\text{st}}$ or ${2}^{\text{nd}}$ century CE, and contains problems involving debts and surpluses that are represented by black and red counting rods, respectively. The text also includes rules for manipulating positive and negative numbers, such as adding, subtracting, multiplying, and dividing them, and finding their square roots.
The Indian mathematician Brahmagupta (${7}^{\text{th}}$ century CE) was one of the first to explicitly treat negative numbers as valid numbers in their own right, and to give rules for their arithmetic, including the product of two negative numbers being positive. He also used negative numbers to represent solutions to quadratic equations, and to denote the direction of motion of celestial bodies. However, he did not accept negative numbers as coefficients in equations, and considered zero and negative numbers as non-numbers or voids in some contexts.

## How can we use a number line to add and subtract negative numbers?

The number line is a really helpful visual tool for working with negative numbers. Whenever we add a negative number, we move left on the number line, and whenever we add a positive number, we move right. For subtraction, we move in the opposite direction! Then the sum or difference is the final number we reach on the number line.
We can also subtract by finding both numbers on the number line. Then the distance between those numbers is the absolute value of the difference. For example, if we have the expression $-7-\left(-9\right)$, we see that $-7$ and $-9$ are $2$ units apart on the number line. So $|-7-\left(-9\right)|=2$.

## Why do we sometimes put parentheses around negative numbers when we add or subtract?

We put parentheses around negative numbers when we add or subtract to help us avoid making mistakes.
For example, if we want to subtract $-2$ from $5$, we can write it as $5-\left(-2\right)$. By putting parentheses around the $-2$, we can see more clearly that we are subtracting a negative number, not just subtracting $2$.
This can be especially helpful when we are dealing with multiple negative numbers in one equation. For example, if we want to subtract $-2$ from $-7$, we can write it as $-7-\left(-2\right)$. Again, the parentheses help us see that we are subtracting a negative number, which will actually make our answer larger.
So, in short, parentheses can help us avoid mistakes and make equations clearer.

## How do the properties of addition apply with negatives?

Addition is really flexible. The commutative property says that we can add numbers in any order and get the same sum. So we can pick which numbers fit together the easiest. The associative property says that we can group the addends in any way and get the same sum.
The commutative property did not work with subtraction.
$\begin{array}{rl}3-5& \ne 5-3\\ \\ -2& \ne 2\end{array}$
That meant we got stuck when we wanted to simplify expressions with subtraction like $-8a+2b-5a$.
One great thing about working with negatives is that we can rewrite all subtraction as addition of the opposite. Now that we're using just addition, we can commute the terms!
$\begin{array}{rl}-8a+2b-5a& =-8a+2b+\left(-5a\right)\\ \\ & =-8a+\left(-5a\right)+2b\\ \\ & =-13a+2b\end{array}$
The key was that the negative symbol moved along with the terms.
Rewriting subtraction as addition of the opposite lets us use the associative property, too. Instead of needing to add and subtract from left to right, we can use whichever groups make the sum easier.
$\begin{array}{rl}5-17+17-2-8& =5+\left(-17\right)+17+\left(-2\right)+\left(-8\right)\\ \\ & =5+\left(\left(-17\right)+17\right)+\left(\left(-2\right)+\left(-8\right)\right)\\ \\ & =5+0+\left(-10\right)\\ \\ & =-5\end{array}$
With practice, we can do the step of rewriting as addition mentally instead of on paper, but make sure that's okay in your class first.