This would be...the Square root of 2 (about 1.414). The numerical value of square root 2 up to 50 decimal places is as follows: At present, the root 2 value is computed to 10 trillion digits. Let the square root of 2 be the ratio p/q, with p and q as co-prime integers (no common factors). Calculate the positive principal root and negative root of positive real numbers. Why? Let us assume that it is, and see what happens. A famous example is the square root of 2, which is roughly 1.4142, and denoted √2. If it is a fraction, then we must be able to write it down as a simplified fraction like this: And we are hoping that when we square it we get 2: or put another way, m2 is twice as big as n2: See if you can find a value for m and n that works! √ 2 = q × q = q 2 If you know derivatives, a simple method is the Newton-Raphson method. (Each of these has been honored by at least one recent book.) Hence p 2 is even. f(x) = x2 - 2 meets that requirement: if x = sqrt(2) then f(x) = 0 Then f'(x) = 2x Let x0 be a first guess at the Square root of 2. Hence p is even. So 2 * 4(square root of 3) = 8(square root of 3). The square root of 2 is the number which when multiplied with itself gives the result as 2. The steps involved in this method are: Also, try this square root calculator which is an online tool that shows root for any given input. The square root of 2 or root 2 is represented using the square root symbol √ and written as √2 whose value is 1.414. But that still gets the same result: both n and m are even. The answer depends on the extent of your knowledge of mathematics. Reduction ad absurdum: a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an absurd result. f(x) = x2 - 2 meets that requirement: if x = sqrt(2) then f(x) = 0 Then f'(x) = 2x Let x0 be a first guess at the Square root of 2. Why? But hang on ... if both m and n are even, we should be able to simplify the fraction m/n. are perfect squares, which gives the whole number when we take the root of them. When you multiply a whole number by a square root, you just put the two together, with the whole number in front of the square root. Also tells you if the entered number is a perfect square. Then we can write it √ 2 = a/b where a, b are whole numbers, b not zero. It actually means that m2 must be an even number. (from Wikipedia). If the square root has a whole number in front of it, multiply the whole numbers together. 1 $\begingroup$ On wikipedia I read about the continued fraction of the square root of 2: $$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{...}}}}$$ The first convergents are $\frac{1}{1},\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29}$. And we say: It is thought to be the first irrational number ever discovered. Square the root, producing p 2 /q 2 = 2, then multiply both sides by the denominator. Hence q is even. Let us assume that it is, and see what happens.. The first thing that you can try is the popular division method. The square root of 2 is a fascinating number – if a little less famous than such mathematical stars as pi, the number e, the golden ratio, or the square root of –1. m/n (m and n are both whole numbers). Viewed 3k times 4. This leaves p 2 equalling 2q 2. Now, how to find the square root of 2 by division method. But there are lots more. Active 2 years, 1 month ago. If you know derivatives, a simple method is the Newton-Raphson method. The square root of 2 is a quantity (q) that when multiplied by itself will equal 2.