2 Times the Square Root of 2: How-to & Steps

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  • 0:02 Square Roots in Real Life
  • 0:58 Iteration
  • 4:11 Checking Your Work
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Lesson Transcript
Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

How do you find 2 times the square root of 2 without a calculator? This lesson will show you how to do exactly that using the process of iteration, or educated guessing.

Square Roots in Real Life

Stephanie belongs the props and set design crew and is working on the school play. One of her projects requires cutting a square piece of plywood diagonally into two triangles that she intends to use for a picture frame.

Stephanie knows the legs of the triangle are going to be two feet long because the piece of plywood she is going to cut measures two feet by two feet. She knows that the diagonals are going to be longer, but she doesn't know by exactly how much because she doesn't have a calculator or a measuring tape. In geometry class, she learned that the length of the longest side of a right triangle is equal to the length of a shorter side times the square root of two. How can she calculate 2 times the square root of 2?

The square root of 2 simply means that a number multiplied by itself equals 2. Let's see how Stephanie can calculate the square root of 2 without using a calculator.

Iteration

One way to calculate the square root of 2 is through a process of repeated educated guesswork. This process is called iteration and involves using the results of the previous guess to inform the next one. In this way, you can get closer and closer to the actual answer until you reach the point where you decide that you are close enough. But where to start?

Let's start by reviewing what we know about the squares of numbers before and after 2. The square root of 1 is 1, so the square root of 2 will have to be larger than 1. The square root of 4 is 2, so the square root of 2 has to be smaller than 2. Therefore, the square root of 2 is somewhere between 1 and 2. Let's split the difference and try 1.5 as our first guess.

1.5 x 1.5 = 1.5(1 + 0.5) = 1.5 + 0.75 = 2.25

As 2.25 is larger than 2, let's make our next guess 1.4.

1.4 x 1.4 = 1.4(1 + 0.4) = 1.4 + 0.56 = 1.96

We're getting pretty close. In fact, our calculations might be accurate enough for many situations where we're using the square root of 2. However, Stephanie really wants her picture frame to be accurately sized, so her estimate of the square root of 2 must also be accurate to two decimal places. Let's try 1.42 next.

1.42 x 1.42 = 1.42(1 + 0.4 + 0.02) = 1.42 + 0.568 + 0.0284 = 2.0164

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