√2 without pressing √ on a Scientific Calculator

Question

You have to calculate $\sqrt{2}$ on your scientific calculator by only pressing these buttons:

[  7  ][  8  ][  9  ][ DEL ][ A C ]
[  4  ][  5  ][  6  ][  ×  ][  ÷  ]
[  1  ][  2  ][  3  ][  +  ][  -  ]
[  0  ][  .  ][×10^x][ ANS ][  =  ]

...which are the bottom-most buttons on it. How can you achieve this by pressing buttons less than or equal to seven times?

• What it means to calculate the square root is to make a difference smaller than $10^{-9}$. Therefore, if your calculator shows 1.414213562 after the final =, the solution is valid. Higher accuracy isn't a problem at all.
• Important: = does NOT count as button presses.
• Your solution cannot assume that a specific value is previously stored in ANS.
• Just for clarification, you cannot use SHIFT and ALPHA as well.

Note: This problem is a sequel to The Blindfold CASIO fx-570EX Puzzle. However, you can use any scientific calculator for this challenge.

Answer 1

I wonder if this is the answer you have in mind, because it feels kind of like cheating... Also depends on the calculator's syntax, so tell me if it's well-typed.

[ 0 ] [ = ] to set the Ans, then, [ + ] [ 1 ] [ x10^ ] [ 0 ] [ - ] [ 9 ] and repeat [ = ] as necessary until you reach your number. If [ 0 ] is not needed, this is only six presses (discounting the hundreds of billions of [ = ] presses); if you need an explicit [ Ans ] in the second expression, it would be eight presses; if [ x10^ ] syntax doesn't enter into its own parenthesized input box, it's entirely sunk.

Edit: A technique that works (in five presses!) even if the previous one doesn't:

[ 0 ] [ = ] to set Ans; then [ + ] [ 1 ] and [ = ] as needed; then [ / ] [ 2 ] (or [ / ] [ 1 ] [ 0 ] to be a little simpler, base-10) and [ = ] as needed.

Also, a hint to others regarding "noble" answers:

I've already tried some "obvious" methods involving (1) continued fractions, and (2) functional fixed-points, but to no avail in seven or less presses. Maybe I missed something, maybe not.

Answer 2

One possibility, if starting with an operation automatically prepends Ans:

Start with 1 =.
Then, do the equation: [Ans] / 2 + 1 / Ans. This costs six presses, since we get the first [Ans] for free.

Now mashing = gives us an arbitrarily precise approximation to √2.

Why does this work?

This is Newton's method. We start with any positive number $x$, and consider $2/x$ along with it. These two numbers multiply to give 2.

If one of them is less than √2, then the other will have to be more than √2 to "make up the difference", and vice versa -- that is, the two numbers $x$ and $2/x$ will be on opposite sides of √2. So if we average them to get a new starting point $x'$, the new pair $(x',2/x')$ will be closer to √2 than the first one.
So, we want to do the following process:
[1] Start with any positive number $x$.
[2] Calculate $2/x$.
[3] Take the average of $x$ and $2/x$.
[4] Repeat steps [2] and [3] until satisfied.

So, what is the average of $x$ and $2/x$? Doing some algebra...

$$\frac{x+\frac2x}2 = \frac x2 + \frac1x$$
And this leads to the calculation I gave, Ans/2 + 1/Ans. Step [1] is just putting 1 into Ans, steps [2] and [3] are taken care of by the second calculation, and step [4] is accomplished by repeatedly pressing [=]. So this method does what we wanted it to!

(Side note: It turns out you only need to press it four times to get within $10^{-9}$ - Newton's method converges fast! See this Math.SE answer for another explanation, and the comments for the exact numbers you get at each point.)

Answer 3

2 × × 0.5 =

This works in some contexts where a double multiplication symbol (often typed as the double asterisk **) can be used to exponentiate.