Square Root Of 289 By Division Method

Finding the Square Root of 289 Using the Division Method

This article will guide you through the process of calculating the square root of 289 using the long division method. This method, while seemingly complex at first, is a straightforward way to find the square root without using a calculator, providing a deeper understanding of the mathematical process. Understanding this method is beneficial for improving mathematical skills and problem-solving abilities. We'll break down the steps clearly and concisely.

What is the Division Method for Finding Square Roots?

The division method, also known as the Babylonian method or Heron's method, is an iterative algorithm that refines an initial guess to progressively get closer to the accurate square root. It's based on the principle of repeatedly averaging a number and its reciprocal to approach the square root. While it may seem long compared to using a calculator, it's a valuable technique to understand the underlying mathematical concepts.

Step-by-Step Calculation of √289

Let's find the square root of 289 using the division method. The process involves grouping the digits in pairs, starting from the decimal point (or the units place for integers), and then systematically calculating the square root digit by digit.

1. Pair the Digits: Group the digits of 289 in pairs, starting from the right: 2 89.

2. Find the Largest Integer Whose Square is Less Than or Equal to the First Pair: The first pair is 2. The largest integer whose square is less than or equal to 2 is 1 (1² = 1). Write 1 as the first digit of the square root. Subtract 1 from 2, leaving a remainder of 1.

3. Bring Down the Next Pair: Bring down the next pair (89) to form 189.

4. Double the Current Quotient and Add a Placeholder: Double the current quotient (which is 1), giving 2. Add a placeholder next to this doubled number (2). We need to find a digit (let's call it 'x') such that (2x) * x is less than or equal to 189.

5. Find the Next Digit: We are looking for a digit 'x' such that (20 + x) * x ≤ 189. Trying different values of x, we find that if x = 7, (27) * 7 = 189. This works perfectly! Write 7 as the next digit of the square root. Subtract 189 from 189, leaving a remainder of 0.

6. The Square Root: Since the remainder is 0, we have found the exact square root. Therefore, √289 = 17.

Visual Representation:

The process can be visualized as follows:

   1 7
   -------
√2 89 | 1
   -1
   -------
    1 89 | 27 x 7 = 189
    -1 89
    -------
        0

Conclusion:

The division method offers a manual approach to calculating square roots, deepening your understanding of the mathematical process beyond simple calculator usage. While it might seem more time-consuming than using a calculator, mastering this technique enhances your numerical skills and provides a valuable tool for mathematical problem-solving. Remember to practice this method with different numbers to solidify your understanding. By working through various examples, you'll become proficient in calculating square roots using the long division method.