A Curious Property Of The Square Root Of 2

One day, while doing some calculations involving the square root of 2, of which you’d most likely find boring, I discovered a little curiosity involving the square root of 2 by chance.

Multiply $\sqrt {2}$ by 1, 2, 3 and so on but drop the decimals from the products. For instance, $\sqrt {2}\times 6$ is equal to 8.485… but you don’t have a need for the decimal part. Thus, $\sqrt {2}\times 6$ in this case is 8.

This has nothing to do with rounding up or down. You just have to drop the decimal part. In technical term, this is called the floor function $\left \lfloor x \right \rfloor$, though you don’t have to worry about that.

The following shows the products of the square root of 2 multiplied by 1 through 25 without their decimal parts.

Write down each answer in a horizontal in a horizontal line as shown in Figure 1.

Fig. 1

Now, you will notice that some numbers are missing from the sequence. Write these missing numbers under the numbers in Figure 1. This is shown in Figure 2.

Fig. 2

Subtract the upper number from the lower number.

Note that the difference in these pairs is 2, 4, 6, 8, 10 and so on.

3 – 1 = 2
6 – 2 = 4
10 – 4 = 6
13 – 5 = 8
17 – 7 = 10
etc.

Posted by Edmark M. Law

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. You can find me on Twitter` and Facebook. My email is edmarklaw@learnfunfacts.com

22 thoughts on “A Curious Property Of The Square Root Of 2”

1. Interesting! I might use this knowledge in my GMAT exam for graduate school.

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2. meandyou792 says:

Very interesting
I also have some things I discovered myself about square roots

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3. Interesting post! I’m a starting blogger who wishes to write on interesting math stuff too, and these are the kinds of post I see myself writing.

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Sqrt(2)*1 does not equal 1 it equals sqrt(2)!

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1. Please reread the instruction. I have never said that √2 is equal to 1. I stated that you multiply √2 by 1, 2, 3, etc. and drop the decimal. So in this case √2 × 1 = 1, instead of 1.414…

This is the floor function, i.e. ⌊ √2 × 1 ⌋ = 1

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5. Interesting. I wonder if this can be shown to be generally true. Can we define a function which would easily give us the sequence of “missing” numbers from the list?

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1. It can be proven through induction.

As for the function for the sequence of missing numbers:

a(n) = floor[n(2+√2)].

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6. I don’t think you’d catch me doing any square roots… unless they are attached to a tree…. 😉

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1. Ha ha! I’m still trying to find X 😉

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