Square Root of 3Date: 5/31/96 at 2:49:5 From: Marie Langley Subject: Square root of 3 My granddaughter wants to know, what is the square root of 3? Thanks, Marie Langley Date: 6/2/96 at 17:46:25 From: Doctor Brian Subject: Re: Square root of 3 Hello- In a rare situation, three of our math doctors have addressed your question! Since the three responses have three very different flavors, you're going to get all of them. Enjoy! The square root of 3 is not a "nice" square root such as saying that the square root of 81 is 9 or the square root of 4 is 2. In fact, it can't even be expressed as a fraction, and if we do it as a decimal, then the decimal "goes on forever" because no finite decimal gives us the precise answer. The number 1.732051 is very close, but when squared, you'll get a number just *slightly* larger than 3. Numbers like this, where there isn't a fraction to describe them, and where the decimals don't end or even repeat, are called IRRATIONAL. Usually you have two options - write 'the square root of 3' as an answer, or round off the decimal to 1.73 or 1.732051 or as many places as you wish. -Doctor Brian, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 6/2/96 at 17:47:51 From: Doctor Tom Subject: Re: Square root of 3 The square root of three is a number that cannot be written down exactly as a decimal. Just as if you try to write 1/3 as a decimal, you can say it's approximately (but not exactly) .3333, or a closer approximation is .33333333, or better still, .33333333333333333333333333333333333333333333333333333. But none is EXACTLY 1/3. You can get as close as you want, however. So, the square root of 3 is about 1.732, or, more accurately, it is about 1.73205080756887729353, or, with more decimal places, you can get more accurate approximations. The square root of three, unlike 1/3, never repeats its decimal representation. If you would like to calculate it yourself, here's an easy way: Make a guess (which we'll call "g"), and then calculate the average of g and 3/g. Make this your first guess, and repeat the process as many times as you want to get an answer as accurate as you want. It works even with a bad first guess. Let's see what happens if my first guess is 1: second guess = 1/2(1 + 3/1) = 2 third guess = 1/2(2 + 3/2) = 7/4 = 1.75 fourth guess = 1/2(1.75 + 3/1.75) = 1.732142857142857142857... fifth guess = 1/2(1.732... + 3/1.732...) = 1.7320508100147275405, which is already accurate to almost 8 decimal places. The next guess will be accurate to almost 16 decimal places, and so on. The reason this works is that if your guess is a little too low, then three divided by your guess must be a little too high, and their average will be closer to the real answer. And vice-versa. Of course, you have to understand some higher mathematics to see why the method is so ridiculously good - that 5 simple calculations, starting from a bad guess, give almost 8 decimal places of accuracy. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 6/2/96 at 17:49:40 From: Doctor Ceeks Subject: Re: Square root of 3 I just wanted to add something regarding the square root of 3. The (positive) square root of 3 is: The unique number which multiplies itself into three... spans the distance between opposite corners of a cubic foot... compares the longer leg to the shorter leg of a 30-60-90 triangle, and is the area of an equilateral triangle with side 2. -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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