Arithmetic

Negative x Negative = Positive? Abstract Proofs

Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.

Negative x Negative = Positive? Concrete Illustrations

One of the more common questions we’ve been asked is, How can the product of two negative numbers be positive? Between this post and the next, I’ll put together many of the answers we have given, starting here with examples from the “real world” (gradually getting more abstract), and next time we’ll look at proofs. …

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Talking About Negative Numbers

Last week we looked at what negative numbers mean; here we’ll consider a number of questions we’ve been asked about the terminology of signed numbers: what “negative” means, and other words for negative numbers. Up, down, and opposite This question from 1998 asks about translating words to signed numbers: Converting Words to Numbers Can you …

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How Real Are Negative Numbers?

This week we’ll look at some Ask Dr. Math questions like, “How can a number be less than zero?” and “Why do we need negative numbers?” We’ll see a number of examples of their use, and how negative numbers make life easier.

Two Word Problems About Factors and Sums

(A new question of the week) A couple recent questions involved factoring numbers, in interesting ways. One involves the volume and perimeter of a block of cubes, and the other involves finding numbers with a given HCF (Highest Common Factor) and sum. Both illustrate thinking through a non-routine problem about factors.

Dividing Fractions: Can You Picture It?

We’ve looked at what it means to multiply fractions, including whole and mixed numbers; now it’s time for division of fractions. We’ll start here with pictures, similar to what we did for multiplication, but a little more complicated. Then next time, we’ll see additional ways to understand why we “invert and multiply”.