Trap √2 by squaring
Squares of guesses give a lower and an upper bound.
Which number is √2?
√2 is the length of the diagonal of a unit square. But which number is it? A ruler does not show it. Let us trap it with squares.
Remember: √2 × √2 = 2. Try the guess 1: 1 × 1 = 1, which is less than 2.
What does this tell us about √2?
What this lesson covers
The idea
Comparing squares gives lower and upper bounds on √2 (1² < 2 < 2², so 1 < √2 < 2); squaring numbers with one more decimal place at a time gives ever closer bounds.
Which number is √2?
√2 is the length of the diagonal of a unit square. But which number is it? A ruler does not show it. Let us trap it with squares.
Remember: √2 × √2 = 2. Try the guess 1: 1 × 1 = 1, which is less than 2.
What does this tell us about √2?
- √2 is bigger than 1
- √2 is smaller than 1
- √2 is exactly 1
Trap it and zoom in
Step the guess along the ruler. The toy squares it. Green: the square is less than 2. Red: more than 2. When green touches red, √2 is trapped between them. Then zoom in.
Closer and closer
Comparing squares gives a lower bound and an upper bound: 1² < 2 < 2², so 1 < √2 < 2. Squaring numbers with one more decimal place each time gives closer bounds.
1.4² = 1.96 < 2 1.5² = 2.25 > 2 1.4 < √2 < 1.5
1.41² = 1.9881 < 2 1.42² = 2.0164 > 2 1.41 < √2 < 1.42
1.414² = 1.999396 < 2 1.415² = 2.002225 > 2 1.414 < √2 < 1.415
Notes
Squaring guesses traps a square root between a lower bound and an upper bound. 1 < √2 < 2 1.4 < √2 < 1.5 1.41 < √2 < 1.42 1.414 < √2 < 1.415
Check yourself
We know 3² = 9 and 4² = 16. So 3 < √10 < ?. What is the upper bound?
Answer: 4
9 < 10 < 16, so 3² < 10 < 4². That gives 3 < √10 < 4.
1.45 × 1.45 = 2.1025. What does this tell us about √2?
1.41² = 1.9881 is less than 2. 1.42² = 2.0164 is more than 2. So 1.41 < √2 < ?. What is the upper bound?
Answer: 1.42
1.42² = 2.0164 is more than 2, so 1.42 is an upper bound. 1.41 < √2 < 1.42.
Rani says: “√2 = 1.4, because 1.4 × 1.4 = 1.96, which is close to 2.” Is she right?
- √2 is smaller than 1.45 — correct. Yes! 2.1025 is more than 2, so 1.45 is too big. √2 is smaller.
- √2 is bigger than 1.45. If 1.45 were smaller than √2, its square would be less than 2. But 2.1025 is more than 2.
- √2 is exactly 1.45. Then 1.45 × 1.45 would be exactly 2. It is 2.1025.
- Yes, 1.4 × 1.4 = 2. 1.4 × 1.4 = 1.96, not 2.
- Yes, 1.96 is very close to 2, so they are equal. Close is not equal. √2 × √2 is exactly 2, but 1.4 × 1.4 is 1.96.
- No. 1.96 is less than 2, so √2 is a little bigger than 1.4 — correct. Yes! 1.4 is only a lower bound. 1.4 < √2 < 1.5.