what type of book is this?

Thank you Davis.

No problem.

It's all on me.

Okay... we'll see about that.

Non fictional idiots.

I believe its fictional. I mean, nonfictional.

Suppose you are asked to find the square root of five to the nearest hundredth thousandth or ten-thousandth. Would your first instinct be to whip out a calculator and punch in the problem? Well, imagine doing problems such as this one without ever having the aid of a calculator. Does that sound impossible? Believe it or not, it isn’t! In fact, what’s even more surprising is that a technique for estimating square roots by hand was established around two thousand B.C., long before the discovery of calculators. This brilliant method is called the Babylonian method, and it was first used by the ancient Babylonians in a part of the world that was once known as Mesopotamia. [new screen] So what exactly is the Babylonian method for estimating square roots? The Babylonian method is an algorithm you can use repeatedly until you find the estimate you’re looking for. Here’s how it works. Suppose you were asked to estimate the square root of x to a given place value. The first thing you must do is determine which two consecutive integers the square root of x lies between. Based on this result, you must then make an initial guess of the value of the square root of x. We will call this initial guess r one. Now, you will take the value of x, the radicand of the square root, and the value of r one and substitute them into the formula one-half the sum of r one plus the quotient x divided by r one. Round the result to the given place value and call this value r two. The value of r two is your new estimate for the square root of x. Take the value of x and the value of r two, and substitute them into the formula one-half the sum of r two plus the quotient x divided by r two. Round the result to the given place value, and call this value r three. Now, the value of r three is your new estimate for the square root of x. Continue this pattern of substituting your new estimates into the formula until you get two estimates in a row that are the same for the square root of x. [new screen] Here’s an example. Let’s use the Babylonian method to estimate the value of the square root of thirty-nine to the nearest hundredth. First we must find the two consecutive integers that the square root of thirty-nine lies between. Thirty-nine lies between the perfect squares of thirty-six and forty-nine. Therefore, the square root of thirty-nine lies between the integers six and seven. Now, we have to make our first guess of the value of the square root of thirty-nine. We could make an initial guess of six point five, but thirty-nine is closer to thirty-six than it is to forty-nine. Therefore, it makes sense to guess a value that is less than halfway between six and seven. We’ll let our first estimate be six point © 2013 K12 Inc. All rights reserved. Copying or distributing without K12’s written consent is prohibited. three, and we’ll call this estimate r one. Now, we’ll use our formula. Substitute six point three for r one, and substitute thirty-nine for x. Remember, x is the value of the radicand. Thirty-nine divided by six point three is about six point two, and six point three plus six point two divided by two is six point two five. Six point two five is our new estimate for the square root of thirty-nine. Let’s repeat the process. We’ll call our new estimate r two. Substitute six point two five for r two and thirty-nine for x in the formula one-half the sum of r two plus x divided by r two. Simplify. Thirty-nine divided by six point two five is about six point two four, and six point two five plus six point two four divided by two is about six point two five. Notice that we now have two estimates in a row that are the same. Since we have two of the same estimate, we can stop. The square root of thirty-nine to the nearest hundredth is six and twenty-five hundredths. [new screen] Here’s one more example. Use the Babylonian method to estimate the value of the square root of seventy-six to the nearest tenth. First, find the two integers that the square root of seventy-six lies between. Seventy-six lies between the perfect squares of sixty-four and eighty-one. Therefore, the square root of seventy-six lies between the integers eight and nine. Seventy-six is closer to eighty-one than sixty-four is. Therefore, it makes sense to guess a value that is a little more than halfway between eight and nine. We’ll let our first estimate be eight point six, and we’ll call this estimate r one. Now use the formula. Substitute eight point six for r one, and substitute seventy-six for x. Seventy-six divided by eight point six is about eight point eight, and eight point six plus eight point eight divided by two is eight point seven. Eight point seven is our new estimate for the square root of seventy-six. Now, let’s repeat the process. We’ll call our new estimate r two. Substitute eight point seven for r two and seventy-six for x in the formula one-half the sum of r two plus x divided by r two. Simplify. Seventy-six divided by eight point seven is about eight point seven. And eight point seven plus eight point seven divided by two is eight point seven. We now have two estimates in a row that are the same, so we can stop. The square root of seventy-six to the nearest tenth is eight and seven tenths.