Squeezing it between two fractions
Root 2 is not a fraction. Can we still say where it lies on the number line?
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Step 1 of 5
- Root 2 lies between 1 and 2.
- Zoom into the tenths.
- It lies between 1.4 and 1.5.
- Zoom into the hundredths.
- It lies between 1.41 and 1.42.
Find a gap, then narrow it
Yes. Root 2 lies between two numbers. These two numbers are the ends of a gap. We can make them come closer and closer. To find root 2 is to find a gap that holds it.
A bigger positive number has a bigger square. So pick a number, square it, and compare with 2.
- 1² = 1 and 2² = 4, and 2 is between 1 and 4. So root 2 is between 1 and 2.
- 1.4² = 1.96 and 1.5² = 2.25, and 2 is between them. So root 2 is between 1.4 and 1.5.
- 1.41² = 1.9881 and 1.42² = 2.0164, and 2 is between them. So root 2 is between 1.41 and 1.42.
Each step makes the gap ten times narrower.
| Gap | Lower² | Upper² | Holds 2? |
|---|---|---|---|
| 1 to 2 | 1 | 4 | yes |
| 1.4 to 1.5 | 1.96 | 2.25 | yes |
| 1.5 to 1.6 | 2.25 | 2.56 | no |
| 1.41 to 1.42 | 1.9881 | 2.0164 | yes |
The same method works for root 3. 1.7² = 2.89 and 1.8² = 3.24, and 3 is between them. So root 3 is between 1.7 and 1.8.
Both ends of each gap are fractions. No fraction squares to 2, so root 2 is never an end. Now try root 4. A gap can end at 2, because 2² is exactly 4.