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How to Find The Square Root on Calculator Program

Reviewed by Calculator Editorial Team

Finding the square root of a number is a fundamental mathematical operation with applications in geometry, algebra, and many scientific fields. This guide explains how to find the square root using a calculator program, including step-by-step instructions, formulas, and practical examples.

How to Find the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 × 5 = 25.

Square Root Formula

For a non-negative real number a, the square root is denoted as √a and satisfies the equation:

√a × √a = a

Key Properties of Square Roots

  • The square root of a negative number is not a real number (it's an imaginary number).
  • The square root of 0 is 0.
  • The square root of 1 is 1.
  • For any positive number, there are two square roots: one positive and one negative.

Note: This guide focuses on real, non-negative square roots. For complex numbers, you would use the imaginary unit i where i² = -1.

Using a Calculator Program

Most modern calculator programs, including scientific and graphing calculators, have a dedicated square root function. Here's how to use it:

Step-by-Step Instructions

  1. Open your calculator program.
  2. Enter the number for which you want to find the square root.
  3. Locate the square root function. This is typically represented by the √ symbol or a dedicated "√x" button.
  4. Press the square root button.
  5. The calculator will display the square root of your number.

Tip: Some calculators require you to press the square root button first, then enter the number. Others allow you to enter the number first, then press the square root button. Check your calculator's manual if you're unsure.

Example Calculation

Let's find the square root of 144 using a calculator:

  1. Enter "144" on the calculator.
  2. Press the √ button.
  3. The calculator displays "12" as the result.

Verification: 12 × 12 = 144, which confirms our calculation is correct.

Manual Calculation Method

If you don't have access to a calculator, you can estimate square roots using the following manual method:

Long Division Method

  1. Write the number as a pair of digits from the decimal point.
  2. Find the largest number whose square is less than or equal to the first pair.
  3. Subtract its square from the first pair and bring down the next pair.
  4. Double the current result and find a digit to append to it that, when the entire number is squared, is less than or equal to the new number.
  5. Repeat until you have the desired level of precision.

Example: To find √23 using this method, you would follow these steps and arrive at approximately 4.796.

Using Approximation

For quick estimates, you can use these approximation techniques:

  • For numbers between 1 and 10, use the following approximations:
    • √1 ≈ 1.000
    • √2 ≈ 1.414
    • √3 ≈ 1.732
    • √4 ≈ 2.000
    • √5 ≈ 2.236
    • √6 ≈ 2.449
    • √7 ≈ 2.646
    • √8 ≈ 2.828
    • √9 ≈ 3.000
  • For larger numbers, you can use the fact that √(a × b) ≈ √a × √b.

Common Mistakes to Avoid

When finding square roots, be aware of these common errors:

1. Forgetting to Check the Input

Ensure the number you're entering is non-negative. Attempting to find the square root of a negative number on a real number calculator will result in an error.

2. Misplacing the Decimal Point

When entering numbers with decimal points, make sure the decimal point is in the correct position. For example, entering 1.44 instead of 144 will give you a different result.

3. Using the Wrong Function

Some calculators have both a square root function and a square function. Make sure you're using the correct one for your calculation.

4. Rounding Errors

Be aware that calculators have limited precision. Very large or very small numbers might show rounding errors in the result.

Frequently Asked Questions

What is the difference between square and square root?
The square of a number is the result of multiplying the number by itself (e.g., 5² = 25). The square root is a number that, when multiplied by itself, gives the original number (e.g., √25 = 5).
Can I find the square root of a negative number?
On a real number calculator, no. The square root of a negative number is not a real number. For negative numbers, you would use complex numbers with the imaginary unit i.
How do I find the square root of a fraction?
To find the square root of a fraction, find the square root of the numerator and the denominator separately, then simplify if possible. For example, √(3/4) = √3 / √4 = √3 / 2.
What is the square root of zero?
The square root of zero is zero, since 0 × 0 = 0.
How can I verify my square root calculation?
Multiply the square root by itself and check if you get back to the original number. For example, if you calculated √16 = 4, verify by checking that 4 × 4 = 16.