Lessons from teaching gifted elementary school students (Part 2)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

A \times A = B

B \times B = C

C \times C = D

If the pattern goes on, and if A = 2, what is Z?

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

green_speech_bubble

Let’s calculate the first few terms to try to find a pattern:

B = 2 \times 2 = 2^2

C = 2^2 \times 2^2 = 2^4

D = 2^4 \times 2^4 = 2^8

etc.

Written another way,

A = 2^1 = 2^{2^0}

B = 2^{2^1}

C = 2^{2^2}

D = 2^{2^3}

Continuing the pattern, we see that Z = 2^{2^{25}}, or Z = 2^{33,554,432}.

If you try plugging that number into your calculator, you’ll probably get an error. Fortuniately, we can use logarithms to approximate the answer. Since 2 = 10^{\log_{10} 2}, we have

Z = \left( 10^{\log_{10} 2} \right)^{33,554,432} = 10^{33,554,432 \log_{10} 2}

Plugging into a calculator, we find that

Z \approx 10^{10,100,890.5195}

\approx 10^{0.5195} 10^{10,100,890}

\approx 3.307 \times 10^{10,100,890}

green line

When this actually happened to me, it took me about 10 seconds to answer — without a calculator — “I’m not sure, but I do know that the answer has about 10 million digits.” Naturally, my class was amazed. How did I do this so quickly? I saw that the answer was going to be Z = 2^{2^{25}}, so I used the approximation 2^{10} \approx 1000 to estimate

2^{25} = 2^5 \times 2^{10} \times 2^{10} \approx 32 \times 1000 \times 1000 = 32,000,000

Next, I had memorized the fact that that \log_{10} 2 \approx 0.301 \approx 1/3. So I multiplied 32,000,000 by 1/3 to get approximately 10 million. As it turned out, this approximation was a lot more accurate than I had any right to expect.

Leave a comment

5 Comments

  1. Tanya

     /  September 8, 2016

    John, can you fix the typo in the equation before the last paragraph? It says, or implies, (2^2)5 because the 5 didn’t get superscripted, when you really mean 2^(25). Threw me off for a bit! Thanks for a great post!

    Reply
  1. Lessons from teaching gifted elementary school students: Index | Mean Green Math
  2. Lessons from teaching gifted elementary school students (Part 5c) | Mean Green Math
  3. Lessons from teaching gifted elementary school students: Index (updated) | Mean Green Math

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