what is the square root of 2?

Question

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Answer 1

So, what is the square root of 2. Let's find out. Let s be the square root of 2. Then s squared, that is s^2 = 2. Lets try something ancient :

(s - 1)(s + 1) = s^2 - 1 = 2 - 1 = 1

s - 1 = 1/(s + 1) and so, s = 1 + 1/(1 + s) which we can regard as a peculiar statement. Look again,

s = 1 + 1/(1 + s)

It looks like a never ending recipe. For the value s which is
stuck in the fraction on the right, let's substitute the value s
which the equation says s is equal to, then the equation becomes,

s = 1 + 1/(1 + 1 + 1/(1 + s)) = 1 + 1/(2 + 1/(1 + s))

We may repeat this and s never disappears from the right hand side of this equation and we get a never ending fraction which looks like this,

s = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 +.........................eqn #

What's interesting about this is that we can use it to give better and better approximations to s and s needs some
approximations because the fraction above is the continued fraction expansion for the square root of 2 and it does not
terminate which makes s an irrational number. Anyway for
the approximations we evaluate longer and longer chunks
of the infinitely long fraction in eqn# to get

1 = 1/1
1 + 1/(2 + ..) = 3/2
1 + 1/(2 + 1/(2 +...) = 7/5
next = 17/12
and next = 41/29
and next = 99/70

and so on forever. After a few of these convergents we get
a good approximation to the square root of 2 if we use

99/70 = 1.4143

We may generate the convergents without adding the fractions all the time by using the rule,

"each numerator is twice the previous numerator plus the
numerator before that and the same rule applies for denominators"

For example, 17 + 2(41) = 99 and 12 + 2(29) = 70

After a few more convergents we arrive at,

1393/985 = 1.4142132 and the actual value of sqrt2

is 1.41421356....

Answer 2

If you have a square 1 cm by 1 cm, the distance from corner to corner will be sqrt(2) cm, which is 1.4142135...

With fractions you can always find a pattern to the decimal form of the number. For example
3/7 = 0.42857142857142857142857142857143..
Notice that 428571 is repeated over and over again.
With the sqrt(2) there is no repeating pattern because there is no exact fraction that can represent it.

Answer 3

Hi,

The square root of 2 is an irrational number meaning that it's not a perfect square.

If you do this operation, you find that it's 1.414 or -1.414 because multiplying two negative numbers together gives you a positive number.

Answer: Approximately 1.414 and -1.414

I hope that helps you out! Please let me know if you have any other questions!

Sincerely,

Andrew

Answer 4

It is the number whose square is 2.
EDIT: very nice piece of information there by the guy who did the ancient method of computing sqrt 2. Thanku.

Answer 5

It is an irrational number. To the nearest hundredth, it is 1.41. But to express the precise value, you should leave it in the form "sqrt(2)."

Answer 6

1.4142