Let's Learn Mathematics

Approximating Square Roots

Introduction

Square roots of non-perfect squares are irrational numbers that cannot be expressed as exact fractions. However, we can approximate their values using calculators or mathematical tables.

A perfect square is a number that is the square of an integer (e.g., 1, 4, 9, 16, 25). Numbers that are not perfect squares have square roots that are irrational numbers (e.g., √2, √3, √5).

Examples of perfect vs non-perfect squares:

\[ \begin{align*} \text{Perfect squares:} & \quad \sqrt{16} = 4 \\ \text{Non-perfect squares:} & \quad \sqrt{17} \approx 4.123 \end{align*} \]

Key Concept

To approximate √N where N is not a perfect square:

  1. Find the nearest perfect squares below and above N
  2. Estimate between these two values
  3. Use a calculator or table for more precise approximation

Using Calculators

Step-by-Step Process

  1. Turn on your scientific calculator
  2. Locate the square root (√) button
  3. Enter the number you want to find the square root of
  4. Press the square root button
  5. Round the result to the required decimal places

Example: Find √7 to 3 decimal places

  1. Press √ button
  2. Enter 7
  3. Press =
  4. Result: 2.645751311
  5. Rounded to 3 decimal places: 2.646

Common Mistakes

  • Forgetting to press the √ button before entering the number
  • Not rounding correctly to the specified decimal places
  • Confusing the square root (√) with the square (x²) button

Using Tables

Step-by-Step Process

When calculators aren't available, we can use mathematical tables to approximate square roots:

  1. Locate the square root table in your mathematical tables book
  2. Find the row for the first two digits of your number
  3. Find the column for the third digit
  4. Read the value at the intersection
  5. For numbers beyond the table range, use the difference columns
  6. For numbers with more than 3 digits, use the mean difference columns

Example: Find √3.75 using tables

  1. Locate row for 3.7 (first two digits)
  2. Find column for 5 (third digit)
  3. Intersection shows 1.936
  4. √3.75 ≈ 1.936

Four-Figure Tables

For more precise approximations, four-figure tables provide additional accuracy:

Structure

Four-figure tables show:

  • Main values (first three digits)
  • Mean differences (fourth digit)

Example

To find √12.34:

  1. Find √12.3 = 3.507
  2. Find mean difference for 4 = 6
  3. Add: 3.507 + 0.006 = 3.513
  4. √12.34 ≈ 3.513

Practice Exercise

Question 1

Use a calculator to find √8 correct to 2 decimal places.

√8 ≈ 2.828427125

Rounded to 2 decimal places: 2.83

Question 2

Use tables to find √15.6 correct to 3 decimal places.

From tables:

√15.6 ≈ 3.949

Question 3

Approximate √50 using the nearest perfect squares method.

Nearest perfect squares:

7² = 49 and 8² = 64

50 is closer to 49 than 64

√49 = 7 and √64 = 8

Estimate: √50 ≈ 7.1 (actual: 7.071)

Question 4

Use four-figure tables to find √23.45 correct to 4 decimal places.

From four-figure tables:

  1. √23.4 = 4.837
  2. Mean difference for 5 = 5
  3. Add: 4.837 + 0.005 = 4.842
  4. √23.45 ≈ 4.8426 (actual: 4.8426)

Question 5

Find √2.5 using both calculator and tables, then compare the results.

Calculator method:

√2.5 ≈ 1.58113883

Tables method:

√2.5 ≈ 1.581 (from tables)

The results are very close, with the calculator providing more decimal places.

Ready for more challenges?

Full Practice Set

Math Challenge

Timed Test in 7 Levels

Test your square root approximation skills through progressively challenging levels

Each level has 20 questions. Need 17 correct to advance.