56 Orders of Magnitude
Problem 17
How old were you when you were one million seconds old? (That’s .)
- Before you figure it out, write down a guess. What’s your gut instinct? About a day? A week? A month? A year? Have you already reached that age? Or maybe you won’t live that long?
- Now figure it out! When was / will be your million-second birthday?
Problem 18
How old were you when you were one billion seconds old? (That’s .)
- Again, before you figure it out, write down a guess.
- Now figure it out! When was / will be your billion-second birthday?
Were you surprised by the answers? People (most people, anyway) tend to have a very good sense for small, everyday numbers, but have very bad instincts about big numbers. One problem is that we tend to think additively, as if one billion is about a million plus a million more (give or take). But we need to think mulitplicatively in situations like this. One billion is .
So you could have just taken your answer to Problem 17 and multiplied it by to get your answer to Problem 18. Of course, you would probably still need to do some calculations to make sense of the answer.
Think / Pair / Share
When is your one trillion second birthday? What will you do to celebrate?
Think / Pair / Share
The US debt is total amount the government has borrowed. (This borrowing covers the deficit — the difference between what the government spends and what it collects in taxes.) In summer of 2013, the US debt was on the order of 10 trillion dollars. (That means more than 10 trillion but less than 100 trillion. If you were to write out the dots-and-boxes picture, the dots would be as far left as the place.)
- If the US pays back one penny every second, will the national debt be paid off in your lifetime? Explain your answer.
- A headline from April 2013 said, “US to Pay Down $35 billion in Quarter 2.” Suppose the US pays down $35 billion dollars every quarter (so four times per year). About how many years would it take to pay of the total national debt?
Here are some big-number problems to think about. Can you solve them?
Problem 19
- Suppose you have a million jelly beans, and you tile the floor with them. How big of an area will they cover? The classroom? A football field? Something bigger? What if it was a billion jelly beans?
- Suppose you have a million jelly beans and you stack them up. How tall would it be? As tall as you? As a tree? As a skyscraper? What if it was a billion jelly beans? About how many jelly beans (what order of magnitude) would you need to stack up to reach the moon? Explain your answers.
Fermi Problems
James Boswell wrote,
Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it.
But math proves this wrong. There is actually a third kind of knowledge: Knowledge that you figure out for yourself. In fact, this is what scientists and mathematicians do for a living: they create new knowledge! Starting with what is already known, they ask “what if…” questio