## What is a Ratio, Really?

A recent question reminded me I hadn’t yet written about the complexity surrounding the definition of ratio (and related terms, like rate and fraction). Here are four questions about the words.

A recent question reminded me I hadn’t yet written about the complexity surrounding the definition of ratio (and related terms, like rate and fraction). Here are four questions about the words.

Last week we looked at how to “cast out nines” to check arithmetic, and touched only briefly on its relationship with modular arithmetic and remainders. Here we’ll look at several explanations of why it works, aimed at different levels of students, with varying levels of success..

This old technique for checking arithmetic is both easy and hard to describe: easy to explain in advanced terms, but hard to explain in elementary terms. We’ll try to do it all here, but a fuller explanation of the “why” will come next week.

In recently discussing Roman numerals, we ran across Egyptian multiplication. An improvement on that method is called the Russian peasant method, and deserves attention.

Have you ever wondered how to add, subtract, multiply, and divide using Roman numerals? On one hand, we’ll give the simple answer that the Romans didn’t actually do what you think; on the other hand, we’ll consider what they actually did.

Roman numerals are very different from the “Arabic” system we use; there is no “place value”. And yet, as we’ll see, the two systems have more in common than you might think.

This is part 2 of a series of extracts from discussions we have had on whether multiplication implied by juxtaposition is to be done before division (which I call IMF, for Implied Multiplication First). Some people write to us claiming that there is one official correct answer. Are they right?

Last week we looked at some basics about zero; now let’s look at whether zero is positive or negative, and then at the topic of the recent comment that triggered this series: whether zero is even or odd.

A recent comment on the site raised questions about zero, beyond what we have discussed in the past about division by zero. Here we’ll look at basic questions about whether zero is actually a number at all, and then about multiplication by zero, which confuses a lot of people.

Last week, we looked at problems involving some number of people making some number of things in some amount of time. In a classic twist on this problem, we’ll now examine several variants starting with “If a hen and a half can lay an egg and a half …”. Can we make sense of half-eggs …