What you’ll learn
Approximate irrational numbers.
Key idea
Positive squares preserve order; a lower approximation must stay below the exact root.
A greatest lower tenth stays strictly below the root; an inverse interval must determine a unique radicand.
Worked example
Enter the exact value or ordered values in the format requested below.
Find the greatest tenth strictly below √20. Enter the numerical lower approximation.
- 4.4²=484/25<20<81/4=4.5².
- Thus 4.4<√20<4.5; 4.4 is the greatest lower tenth, while 4.5 is above the exact root.
Answer: 4.4
Quick Check
Evaluate a worked attempt at approximate irrational numbers, identify the first invalid step, and provide the correction.
Step 1 bounds √63 between 7.9 and 8.0 by exact squares. Step 2 chooses 8.0 as the greatest lower tenth because it is larger. Identify the first invalid step and correct it.