Squeezing it between two fractions

Root 2 is not a fraction. Can we still say where it lies on the number line?

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.

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.

Try the lab to finish this lesson.