No fraction fits

About this video

Fold a square napkin with sides of 1 m from corner to corner. How long is the fold? This animation finds out. The square drawn on the diagonal is made of four halves of the small square, so its area is 2, just as Pythagoras' theorem says: 1 × 1 + 1 × 1 = 2. The diagonal is the side of a square of area 2, so it is the number that times itself makes 2: the square root of 2, written √2.

Then the hunt begins. 1.4 × 1.4 = 1.96 is too small and 1.5 × 1.5 = 2.25 too big, so √2 is between 1.4 and 1.5. 1.41 × 1.41 = 1.9881 is too small and 1.42 × 1.42 = 2.0164 too big, so it is between 1.41 and 1.42. A number line zooms in on the gap. Each new digit gets closer, but no decimal ever lands exactly on 2, and the digits of √2, 1.41421356..., go on for ever without repeating.

Fractions do no better: 7/5 times itself is 49/25, just under 2, and 17/12 times itself is 289/144, just over. Greek mathematicians proved more than 2,000 years ago that no fraction at all, times itself, makes exactly 2. Yet √2 is a real length: swing the diagonal down onto the number line like a door and it lands at its own exact place, between 1.41 and 1.42. A number like this, which is not a fraction and whose decimals never end and never repeat, is called an irrational number. This video goes with the A Length No Fraction Can Name lesson of Geometry Level 1 on Knaph.

Transcript

1. 0:00A square napkin, one metre each side. How long is the fold from corner to corner?
2. 0:08The square on the fold is four halves of the napkin. Its area is two.
3. 0:17So the fold is the number that, times itself, makes two. The square root of two.
4. 0:24Let's hunt for it. One point four is too small. One point five is too big.
5. 0:33One point four one, too small. One point four two, too big. We get closer and closer, but never land on two.
6. 0:44Its decimals go on for ever, and never repeat.
7. 0:52Fractions don't work either. Seven fifths is just too small. Seventeen twelfths, just too big. No fraction is exactly right.
8. 1:02Yet it is a real length. Swing the fold down onto the number line. It lands at its own place.
9. 1:14A number no fraction can name is called irrational.