When we discover negative numbers we naturally, without question even, assume they obey the same laws of arithmetic as the ordinary positive counting numbers. In the early curriculum multiplication is introduced in the context of whole counting numbers and is appropriately defined there as repeated addition.
Quick Insight: Subtracting Negative Numbers
The computation is quite different with the numbers 5 and 4 serving switched roles. Question: How would you convince a young student that groups of is sure to have the same value as groups of ? HINT: Put dots in rectangular arrays.
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Repeated addition allows us to multiply a positive number and a negative number. Using piles and holes this looks like:. Interpreting negative times a positive and negative times negative through repeated addition, however, is problematic. The truth is that multiplication has no meaning here in context of repeated addition. We have entered new territory and if we want to open up our world to new types of numbers it is not surprising that previously concrete, literal definitions begin to flail.
So we have to engage in a sophisticated shift of thinking, letting go of the question What is multiplication? How would we like multiplication to behave? In other words, students should know that:.
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These numbers are called opposites or additive inverses. Putting this together with the fact that depositing money in the bookstore account means adding an amount to the account balance, and buying an item from the bookstore means subtracting an amount from the account balance leads us to see why it makes sense to represent debt with negative numbers.
Negative number - Wikipedia
It makes sense to use a negative number to represent Evan's account balance when he owes money. Here are two ways to look at it:.
follow url If he spends more than he has in his account, then we are subtracting a bigger number from a smaller number, and the result is negative. If you add two numbers together to get 0, then the numbers must be opposites. The opposite of a positive number is a negative number, so it makes sense to represent his account balance with a negative number when he owes money to the bookstore.
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Alignments to Content Standards: 7. If positive represents movement to the right, negative represents movement to the left. If positive represents above sea level, then negative represents below level.
If positive represents a deposit, negative represents a withdrawal. They are often used to represent the magnitude of a loss or deficiency.
A debt that is owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative.
The laws of arithmetic for negative numbers ensure that the common sense idea of an opposite is reflected in arithmetic.