Negative Numbers

Background

We've explored subtraction when we have a number that is smaller being subtracted from a number that is bigger, such as 3 - 2 (which equals 1). But what happens when we reverse the order and subtract a bigger number from a smaller number?

Question

What's 2 - 5?

Answer

2 - 5 = -3

Analysis 

What does happen when we start with a number and then subtract from it something bigger? That has been a question that has plagued mathematicians, especially ones in Europe, for a long long time.

For the keepers of tax records in countries such as China and India, they had mechanisms to keep track of this:

"You owe 10 chicken eggs to the Overlord. You have 6 eggs. Therefore you owe another 4 eggs to the Overlord."

The Chinese record keepers, starting at least in 200 BCE, kept track of these kinds of debts by using black and red marks: black being payments and red being what was still owed.

In India, starting around 620 CE, the mathematician Brahmagupta devised rules regarding what he called "Fortunes" and "Debts" - or in other words positive and negative numbers.

But as late as the 1750s, European mathematicians were still struggling with negative numbers as a concept. It was only in the 1800s that the rules of arithmetic were formalized and negative numbers were no longer controversial.

One of the ways this controversy ended was that the number line was invented (that's right - the idea of laying out numbers in a row didn't happen until the 1600s) and using the logic of the number line in a positive direction, the negative direction could be better perceived.

The number line, just like it has numbers increasingly positive and larger in value as we move from 0 to the right, also has numbers that are increasingly negative and smaller in value heading to the left. It looks like this:



We can move along the number line using the same rules that we've discussed in prior entries.

Let's work the 2 - 3 problem using the number line and see that it works. We start with an orange dot on the 2:



We now move 5 spots to the left (we can do that in purple):




For more information check out these links (comment to add your favourite link):

The History of Negative Numbers, at nrich.maths.org

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 0
• Whole Numbers

Operations:

• Subtraction

Graphing:

• Number Line

Where might we go?

Numbers:

• Integers

Operations:

• Multiplication
• Division

Associated Operations:

• Absolute Value

Properties:

• Commutative Property