I came across the complying with YouTube video clip a while earlier which supplies a weird trick come accurately almost right square roots. Ns suggest watching at least the an initial minute where the presenter describes how it’s done:

Let’s execute an example. Approximate to 2 decimal places: 

using this method. The following perfect square is 100, whose root is 10. Together for the fraction, the molecule is the difference in between 100 and 96, and the denominator is 2 time 10. So we have:


Ok, there room still peaks, however these are whereby the forward and also backward approximates intersected, so notice how little the error worths are contrasted to earlier.

This is where I end my analysis, and with this hybrid method you have the right to make fairly precise psychological estimates of arbitrarily square roots, however one who is familiar with this errors can conveniently improve their approximations if lock want. Notice how all these errors are positive, i.e. The calculation is always greater than (or equal to) the really value. Also, the the peaks the the hybrid estimate are currently in the middle of regions between perfect squares. Therefore in order to enhance your guess, girlfriend could subtract something small (usually 0.01 or 0.02) when square rooting a number in the “middle regions” in between perfect squares come account for the supposed error. If we’d excellent this with the an initial example earlier, we might’ve got the answer spot on. It’s a little bit of a thumb-suck, due to the fact that it depends on wherein you space on the number line and also how “close” you room to the peak, but this is not an exact science, so thumb-suck away.

Finally, the vital question: why walk this also work? The answer is: since we’re secretly taking a linear approximation the the square source function.

You are watching: How to do square roots in your head

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In various other words, we’re drawing the tangent heat to the square root curve at a point, and also then approximating it’s neighbors using the tangent line instead of the curve. To be an extremely clear:


We could use it come any duty we like, but for now we’re only interested in square roots:



Ring a bell? If 

is what we’re rooting, and 
is a nearby perfect square, then this is just a mathy way of saying what our YouTube girlfriend was telling us from the beginning!