## Mathematical Inconsistencies

There are several cases where my explanations contain mathematical inconsistencies which seem to be unavoidable. Here are a few examples.

### Example 1

In explaining how to divide \(345\) by \(4\) using conventional division, I start out by saying "You can't divide the \(3\) by \(4\), so you put a zero over the \(3\) and ... etc., etc."

But then when explaining how to convert \(\frac{3}{4}\) to a decimal number, I say "You simply divide the \(3\) by \(4\)".

In the one case I say you can't divide \(3\) by \(4\), in the other case I say you can.

### Example 2

When explaining how to subtract \(4\) from \(53\), for example, I say "You can't subtract \(4\) from \(3\) so you have to borrow ... etc., etc."

But then in algebra you subtract \(4\) from \(3\) and get \(-1\).

First you can't subtract \(4\) from \(3\), then later you can.

### Example 3

In fractions, \(\frac{3}{4}\) is not the same as \(\frac{4}{3}\). But in ratio word problems, for example, \(\frac{3\text{ oranges}}{4\text{ dollars}}\) is equivalent to \(\frac{4\text{ dollars}}{3\text{ oranges}}\).

In the one case you change the value of a fraction if you invert it, in the other you don't change the value by inverting it..

(The rationale in this case is that for the fraction \(\frac{3}{4}\), both the \(3\) and the \(4\) are presumed to have the same units (or no units) so that the ratio is dimensionless. When the \(3\) and the \(4\) have different units, oranges and dollars in this case, the fraction is not dimensionless and reflects the relationship between oranges and dollars.)

John Schwarz