A length you can draw and cannot write

Draw a square with sides of 1 unit. Draw its diagonal, the line from one corner to the opposite corner. What number is that length?

Its square is exactly 2

The diagonal is √2 units, the positive number whose square is 2. We say it squares to 2.

Draw a square on each side of a right triangle. The two small squares add to the big one. Both legs, the two short sides, are 1. Each small square has area 1 × 1 = 1. So the big square has area 1 + 1 = 2.

The diagonal is a side of the big square. Call that length the square root of 2, written √2. A root names a side from its area.

No fraction squares to exactly 2

Try 7/5. Its square is 49/25. And 2 is 50/25. The square misses 2 by 1/25.

Try 99/70. Its square is 9801/4900. And 2 is 9800/4900. The square misses 2 by 1/4900.

Three misses do not show that every fraction misses. One argument can show that none works. A number that is not an integer over an integer is irrational. It is not a ratio.

Try the lab to finish this lesson.