The Square Root Demystified: Instantly Understand Why √16 = 4!

The concept of the square root is fundamental in mathematics. In this blog post, we will explore what square roots are, how to calculate them, and their applications in various fields. This guide is designed for beginners and will cover both basic and advanced topics.

ⓘ This post was written after studying with lots of information like books, internet sites, etc. Except in some cases where it is specifically stated, images are made by R & Python using RStudio or VS code.

What is a Square Root?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9. Mathematically, this is represented as:

Graph by Python(the source code is available at the bottom of this post)

Properties of Square Roots

Understanding the properties of square roots can help in simplifying complex calculations:

• The Principal Square Root: Every positive number has two square roots, one positive and one negative. The principal square root is the non-negative one. For example, the square roots of 9 are 3 and -3, but the principal square root is 3.
• Square Root of Zero: The square root of zero is zero itself. This is because 0 × 0 = 0
• No Real Square Roots for Negative Numbers: Negative numbers do not have real square roots because squaring any real number (positive or negative) always results in a positive number. The square roots of negative numbers fall into the realm of imaginary numbers, represented using the imaginary unit “i” (where i² = -1).

How to Calculate Square Roots

1. Using Prime Factorization

Break down the number into its prime factors and pair them. For example:

The remaining 2 doesn’t have a pair, so it stays under the radical

Pros:

• Easy to understand conceptually.
• Works well for smaller numbers.

Cons:

• Can be difficult for larger numbers with many prime factors.

2. Long Division Method

This method is ideal for calculating the square roots of larger numbers without a calculator.

Example: Finding the Square Root of 1849

1. Group Digits: Starting from the right, group the digits in pairs: 18 49.
2. Find the Largest Square: Find the largest perfect square less than or equal to the leftmost group (18). In this case, it’s 4 (4² = 16).
3. Subtract and Bring Down: Write 4 as the first digit of the result. Subtract 16 from 18, leaving 2. Bring down the next pair of digits (49) to get 249.
4. Double the Quotient: Double the current quotient (4) to get 8.
5. Find the Next Digit: Find the largest digit ‘x’ such that 8x times x is less than or equal to 249. In this case, x = 3 (83 x 3 = 249).
6. Subtract and Repeat: Write 3 as the next digit of the result. Subtract 249 from 249, leaving 0.

Since there are no more digits to bring down, the SR of 1849 is 43.

Pros:

• Can be done by hand for precise results.

Cons:

• Can be a bit tedious for very large numbers.

3. Using a Calculator or Computer Program

This is the quickest and easiest way to find square roots, especially for large or complex numbers. Most calculators have a square root button (√), and you can also use programming languages like Python (math.sqrt()) or online calculators.

Example:

• Calculator: Input 16 and press the √ button to get 4.
• Python: math.sqrt(16) returns 4.

Pros:

• Fast and accurate.
• Works for any size number.

Cons:

• Requires a calculator or computer.
• Doesn’t help you understand the underlying process.

Which Method Should You Use?

• Small Numbers: Prime factorization is often the easiest.
• Large Numbers: Use a calculator or computer program for speed and accuracy.
• Practice: If you want to learn the long division method, give it a try for practice!

Conclusion

The square root is a versatile mathematical concept with wide-ranging applications. Whether you are solving simple equations or working on complex engineering problems, understanding square roots is essential. Practice different methods of calculation to improve your proficiency. For more in-depth information, visit to wikipedia.

Why don’t you learn exponential funtions! Check the button below.

# Python Code for Graphs

Here’s the Python code to generate the graphs:

import matplotlib.pyplot as plt
import numpy as np

# Set the range of x values
x = np.linspace(0, 10, 100)

# Compute y = sqrt(x)
y = np.sqrt(x)

# Create a figure object
fig, ax = plt.subplots(figsize=(6, 4))

# Plot the graph
ax.plot(x, y)

# Mark the point (9, 3)
ax.plot(9, 3, 'ro', markersize=8)
ax.text(9.2, 3.1, '(9, 3)', fontsize=10)

# Set the title and axis labels
ax.set_title('Square Root Graph')
ax.set_xlabel('x')
ax.set_ylabel('y = sqrt(x)')

# Display the grid
ax.grid(True)

# Save the graph
plt.savefig('square_root_graph.png', dpi=300, bbox_inches='tight')
plt.show()

Would you like to run the Python code and see the graph directly? If so, please click the button below to install VS Code first. It is explained in Korean, but you can right-click on the post body to translate it into English.