0.999... = 1

Maybe this mathematical curiosity is more widely appreciated than I realize. However, I only recently became aware of it, so I will share with you the madness that I've discovered.

It turns out that 0.999... (the “...” means that the 9s go on forever) is equal to 1. And I don't just mean that the two numbers are extremely close to one another or that they are equal by convention. Rather, I'm telling you that the symbols 0.999... and 1 represent the exact same real number and that this equality can be proven.

If you're learning of this for the very first time, I can guess how you are feeling:

I was feeling the exact same way, and I knew what had to be done. Research mode engaged.

After about 20 minutes of intense research (e.g. Wikipedia, various math blogs, etc.), I was entirely satisfied that 0.999… = 1.

Preliminaries

First off, Wikipedia has an entire page on 0.999… = 1. If that doesn’t settle the matter, then I don’t know what will.

Next stop WolframAlpha, a “computational knowledge engine” that knows pretty much everything. For example, WolframAlpha knows the estimated number of atoms in the sun (9 x 10^56), the average undergraduate tuition at Harvard ($33,700 US), the population of Bahrain in 1962 (161,000 people), and even the phase of the moon on the night that Leonardo da Vinci was born (waning crescent). WolframAlpha also knows that 0.999… is equal to 1.

Evidence

Of course, the above arguments are really only appeals to authority, not actual evidence. So let’s get down to business.

Hopefully we can all agree that

1/9 = 0.111…

If so, we can also agree that

9 * 1/9 = 9 * 0.111...

We’ve simply multiplied both sides of the equation by 9, so the equality still holds true.

If we now simplify both sides of the equation, we are left with

1 = 0.999…

Crazy, right?! Another way to think about this is to ask: is there any number that can fit between 0.999… and 1? If two numbers are truly different, then at least one other number should fit in between them. But nothing fits between 0.999... and 1. For example, if we subtract 0.999... from 1, we get

1 - 0.999... = 0.000...1

The “...” on the right side of the equation represents an infinite number of 0s, so the 1 at the end is irrelevant. Subtracting 0.999... from 1 leaves us with nothing – a big fat goose egg.

You might have guessed that this strange phenomenon is not limited to 0.999… and 1. For the same reasons described above

1.999... = 2
0.42999... = 0.43
99.999... = 100, etc.

In fact, every nonzero number that ends in an infinite number of 0s (e.g. 1 can also be written as 1.000…) has a counterpart that ends in an infinite number of 9s.

See here and here for more details on this mathemagical weirdness.

c²… c² run

a² + b² = c², and that is a fact! At least I think it is. That’s what I learned in grade 8 anyway, so it’s probably true, especially since my grade 8 teacher taught the theorem to our class via song and interpretive dance. People rarely tell lies through interpretive dance, so I had no reason way back in grade 8 to doubt the validity and universality of the Pythagorean theorem.

In my old age, however, I am crusty and skeptical and disinclined to take Pythagoras’ (and Mr. Baumgartner’s) word as established truth. I recently decided to embark on a quest to figure out whether there is any evidence in support of the Pythagorean theorem. What follows, in brief, is what I found…

Is there any evidence to support the Pythagorean theorem?

Yes. Boatloads. There are hundreds of published mathematical proofs of the Pythagorean theorem. Check out this excellent page for 92 of them.

What exactly does the Pythagorean theorem state?

The theorem states that for any right triangle, the squared length of the hypotenuse is equal to the sum of the squared lengths of the other two sides, commonly written as: a² + b² = c².

This equality can be thought of in terms of areas of squares build on each side of a right triangle, as illustrated in the figure below. Because the area of a square is simply the squared length of any of its sides, the Pythagorean theorem essentially states that the area of the square on the hypotenuse (yellow square) is equal to the sum of the areas of the squares on the other two sides (blue and pink squares).

Prove it...

Here is one proof that involves simple geometry and just a splash of algebra:

Start with 4 identical copies of a right triangle.

Rotate the triangles to 0°,  90°, 180°, and 270°, respectively.

Form these triangles into a square such that each side of the large square is the hypotenuse (C) of the original triangles. Note that the area of this large square will be C².

The large square has a square hole inside of it. The sides of this small square are equal to A - B.

The area of the large square (C²) must equal the summed areas of the constituent shapes – the 4 triangles and the small inner square.

The area of each triangle (½base • height) is ½AB, and the area of the small inner square is (A-B)².

So we have:

C²  =  4(½AB) + (A - B)²

C²  =  2AB + (A² - 2AB + B²)

C²  =  A² + B²

Hurray!

Was Scarecrow correct?

No. In The Wizard of Oz (1939), Scarecrow demonstrates his ‘knowledge’ by claiming “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh joy! Rapture! I’ve got a brain!”

Not even close, Scarecrow. You can watch this epic math fail here via YouTube.