The long side as a decimal
A right triangle has short sides 1 and 2. Its long side has c × c = 5. What is c as a decimal?
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c < 3. Step 1 of 4
- 2 × 2 = 4 and 3 × 3 = 9. So 2 <
c< 3. - 2.2 × 2.2 = 4.84 and 2.3 × 2.3 = 5.29. So 2.2 <
c< 2.3. - 2.23 × 2.23 = 4.9729 and 2.24 × 2.24 = 5.0176. So 2.23 <
c< 2.24. - 2.236 × 2.236 = 4.999696 and 2.237 × 2.237 = 5.004169. So 2.236 <
c< 2.237.
The rule
c is between 2.236 and 2.237.
The number with c × c = 5 is written √5. We read √5 as "square root of 5". No short decimal squares to exactly 5. So we find two numbers it lies between.
Rule: Square your guess. Too low, guess higher. Too high, guess lower.
Too low means the square is below 5. Too high means the square is above 5.
Keep one guess that squares below 5. Keep one that squares above 5. Each new digit makes the gap between them ten times smaller.
The digits of √5 go on for ever. No fraction equals √5. The number √5 starts 2.2360679.