Each number system was built to answer a question the one before it could not. Subtraction forces negative numbers, division forces fractions, and one further step is still needed. The fractions look complete, since between any two of them lies another, yet there are points on the line no fraction reaches.

Picture a ruler marked at every fraction. The marks crowd together so tightly that no gap is visible anywhere. Even so, the diagonal of a unit square, laid against that ruler, ends between the marks and not on one.

Each system sits inside the next: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$. Every ring was added to answer a question the one inside it could not — subtraction, then division, and then the gaps this chapter ends on.

Each system contains the one before it and adds what that one lacked. The last step is the subject of this chapter, and the first lesson only shows why it is needed.

• $\mathbb{N}$ counts. It has no answer to $2 - 5$.
• $\mathbb{Z}$ adds negatives, so subtraction always works. It has no answer to $1 \div 2$.
• $\mathbb{Q}$ adds fractions, so division by anything but zero works. It has no answer to .

Suppose such a fraction existed and write it in lowest terms. Evenness then spreads from the numerator to the denominator, so the fraction was not in lowest terms after all. This is the proof by contradiction from the previous chapter, put to work.

Assume the opposite of the claim, and insist the fraction has been fully cancelled.

Square both sides and multiply through, which removes the fraction.

Twice something is even, and only an even number has an even square.

Write the evenness as an equation and substitute it back.

The same argument now applies to the denominator.

A common factor of two was supposed to have been cancelled already, so the assumption fails.

$2 - 5$ has no answer in $\mathbb{N}$ but does in $\mathbb{Z}$.

Subtraction escaping the counting numbers is exactly what the negatives were added for.

Between $\tfrac{1}{2}$ and $\tfrac{2}{3}$ lies .

The trick works between any two fractions, so no smallest gap exists. Gaps of a different kind remain.

$\sqrt{2} \in \mathbb{R}$ but $\sqrt{2} \notin \mathbb{Q}$

It sits on the line and can be constructed with ruler and compass, yet no fraction equals it.

Evenness spreads from one to the other, and a fully cancelled fraction cannot have both parts even.

One combination is forced: adding a rational to an irrational can never give a rational.

• $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$, each built to answer a question the one before could not.
• No fraction squares to two, and the proof is by contradiction from lowest terms.
• The irrationals are not closed under addition or multiplication.
• Density is not the same as having no gaps, and the difference is this chapter.

Great work! You now know how to:

• Say what each number system adds to the one before it
• Run the proof that no fraction squares to two
• Tell density apart from having no gaps