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?
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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.
| Fraction | Its square | 2, same bottom | Misses by |
|---|---|---|---|
| 7/5 | 49/25 | 50/25 | 1/25 |
| 17/12 | 289/144 | 288/144 | 1/144 |
| 99/70 | 9801/4900 | 9800/4900 | 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.
Watch out!
1.414 is not √2. 1.414² is 1.999396, not 2. And 1.414 is just one more fraction that misses.