Square Complex Numbers In Your Head Quickly

Square Complex Numbers In Your Head Quickly

Quick: What’s 19²… without your calculator? (361.) Squaring complex, two-digit numbers in your head isn’t that difficult if you know a simple trick, according to the Secrets of Mental Math author Michael Shermer.

Photo by farnea.

Arithmetic doesn’t always have to be done from right-to-left — in fact plenty of other people do it the other way around. If you see a problem like 32+18, adding it in your head from left to right lets you see 40 first (30 + 10), and then the 10 (2 + 8). A similar process that can be used for multiplication. But what if you want to square numbers like 33²? It’s a little bit more complex, but not so much more so:

[W] hat’s 27². Of course, the brute force way to do that is to calculate 27 x 27 which is a bit of a pain because it involves doing something like 27 x 20 + 27 x 7 = 540 + 189 = 729. But there’s a much faster way.

Observe that 27² = 30 x 24 + 3². Since you probably know that 32 = 9 this means you have to calculate 30 x 24 + 9 which is relatively easy because the multiplication involves a multiple of ten which means it’s really 3 x 24 and then add a zero.

So the rule is that if you want to square number X you first round it to the nearest multiple of 10, called that X + r, and then calculate X – r (i.e. round the same amount in the opposite direction). You calculate (X + r) x (X – r) and add back the square of the amount you rounded by, r², which will be 1, 4, 9, 16 or 25.

This works because ( X + r ) x ( X – r ) + r² = X² – rX + rX – r² + r² = X².

I’m adding this tip to my mental repertoire, right next to my mental trick for quickly calculating a 20 per cent tip quickly when eating out (move the digit over one place to the left, and double it). If you’ve got any useful maths tips for the rest of us who haven’t taken maths classes in years, we’d love to hear it in the comments.

Squaring two digit numbers in your head [John Graham-Cumming]


  • are you trying to make me an april fool? First took me a few seconds to figure out how 9=32 (should be 3^2) and then I thought i’d try my new squaring skillz to get the same answer as you for 19^2 and it was 361 not 351!

  • Please change the misleading title. This is not how to square complex numbers at all. The word complex should not be used in this context when you are referring to integers.

    • Yes. Complex numbers consist of both a real and an imaginary part. Additionally, multiplying two-digit numbers mentally can hardly be considered complex. Additionally, the article covers how to multiply any two two-digit numbers mentally, not just how to square them.

      Make the kids who did the hardest maths in high school happy.

  • Not to be pedantic, but a complex number in mathematics is defined as the square root of a negative number. So, sqrt(-3) is a complex number and when you square it, the answer is ….. -3.

  • I agree with Dan.

    Complex numbers means something totally different. eg. 4 + j2

    The more appropriate title would be Square Large Values In Your Head Quickly.

    Some other tricks I use are if you multiply something by 5. Times 10 then divide by 2.

    another trick I remember: multiply 2 some what consecutive numbers: eg. 299 * 301

    the trick is you square the middle number between the two numbers an subtract 1.

    so 300 * 300 – 1 = 90,000 – 1 = 89,999

  • For those who know how to pick apart an article – who cares ! If you are so smart, perhaps you could actually put in a useful tip as requested.

    Here is one for you…
    If you want to multiply a double digit number by 11, add the two numbers together and put it in the middle. If the middle number is >10, carry the 1.

    eg. 23×11 = 253 (2) (2+3) (3)
    32×11 = 352 (3) (3+2) (2)
    53×11 = 583 (5) (5+3) (3)
    82×11 = 902 (8) (8+2) (2) (carry the 1)

  • To be even more pedantic, any number can be considered a complex number – it’s just that a lot of them have Im(z)=0 (that is, z=a+0i). Real numbers are simply a subset of complex numbers, hence that property.
    Then again, this article has nothing to do with squaring all complex numbers. Nonetheless, it is an interesting trick.

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