Calculators: Accurate, Fast, and Often Wrong!

I was visiting with a friend of mine this past Friday. The love of her life, her only grandchild, was staying with her for the weekend. He’s either just entering seventh grade or eighth grade; I’m not sure which.

Anyway, he had decided to do his homework assignments right away—starting with math—so he’d be free to fully enjoy the rest of his weekend.

Now, for the record, I MUST digress a bit. Let it be known that I’ve NEVER operated this way. Yes, I admire such people, but I consider them structured wussies.

I ALWAYS waited until the last minute to begin my homework assignments, and I did so under the rationalization that I always performed at my best while under a LOOMING iron-fisted deadline that spelled catastrophe had I missed it.

Nothing has changed, either. I’ve been writing this blog since I retired at the beginning of January 2003. Exclusive of this one, I’ve published 555 articles to date. And for each of them, including this one, I’ve had a full week to prepare them.

But with the exception of five postings, I’ve not taken anywhere near a week to write anything I’ve posted. It’s not that I’ve never started out at the beginning of each week with a planned topic; it’s just that something has always come up late in a given week, and I’ve inevitably ended up at my laptop around 10:30 AM on Sunday starting from scratch.

So here I sit once again writing from scratch about something that didn’t even happen until late this past Friday afternoon. So now that you know this, I’ll get back to my friend’s grandson and what seemed like a bit of confusion regarding his math assignment.

Math has been an integral part of my life over a 40-year career in physics and engineering. But since my own children are grown and well into their own lives and careers, it’s been a LONG time since my last contact with seventh and eighth grade math. But the point is that you don’t forget the stuff!

In talking with this youngster, it was evident—at least it was to me—that he was quite able-minded both intellectually and emotionally. But he also was legitimately confused over exponential notation.

He was comfortable with the concept. However, the confusion arose over the difference between the way his teacher explained exponential notation and the way scientific calculators actually display it.

This isn’t a math lecture, so I’m not going into a lot of boring details here. Suffice it to say that exponential notation is science’s way of stating very large and very small numbers.

For example, one could write out the number 1,000,000. Or one could state the number as 106. The EXPONENT (6) tells how many zeroes there are to the right of the 1.

If you wanted to multiply 10,000,000,000 (1010) by 100 (102), all you’d have to do is add the two exponents and state the answer as 1012. But if you’re obsessed with writing out large numbers, all you’d have to do is write a 1 followed by 12 zeros (1,000,000,000,000).

But when we use scientific calculators to do exponentiation, we must grasp the difference between the reasoning powers of the human mind—although slow—and the immense speed of scientific calculators—although incapable of reasoning.

All hype aside, computers and scientific calculators are only as exact as their programming dictates. They’re incredibly fast, but stupid. Humans, by all comparisons, are slow to a fault, often inaccurate, at times brilliant, and at other times abjectly stupid.

But make no mistake about it; combining the two can be incredibly powerful. But whether the power turns out positive for the well-being of the human race or negative, even species-ending, will ultimately depend entirely on the common sense of HUMANS, not computers.

Scientists the world over know that when they see a 10 with an exponent, the exponent determines the number of zeros that follow the 1. But when you use a scientific calculator, the result depends on which exponentiation key one uses.

If you enter a 10, tap the EXP (on some calculators it’s EE) and enter 5, then tap =, the result will be 1,000,000 because this function interprets the 10 as followed by 5 zeros.

Using the same calculator, entering 10, then tapping the XY (some calculators use YX) followed by 5 and the = key, the result will be 100,000 because this function interprets a 1 followed by 5 zeros.

But either way, the HUMAN operator has to know what’s going on and compensate accordingly. But often they don’t, and this includes some of the teachers.

People who simply hit calculator keys while blindly accepting the results as Biblical revelations should never be allowed anywhere near a calculator or a computer.

But more important than the mechanics involved in exponential calculations, it’s far more important to understand their implications. This involves a firm conceptual understanding of exponentiation!

For example, using world population annual growth rates from 1910 through 2010, the average growth rate was 1.5%; or stated another way, the world population grew at an exponential rate of 1.5% over 100-years.

While this rate won’t continue unabated in any way, what would actually happen if it did remain a constant?

We’re at about 7-plus billion people right now. If you multiply 7-billion by (1.015)50, you’ll see that the world population will approximately double over the next 50-years.

Here’s an even neater example of the power of exponentiation. Suppose that Bill Gates offers to toss money into your backyard for a 31-day period. On the first day, he tosses in 1-penny. On day two, he doubles it to 2-pennies. On day three he doubles it again to 4-pennies.

In other words, he starts with 1-penny, then 2, 4, 8, 16 etc., right on through day 31. He’s exponentially doubling the amount he gives you each day for 31-days.

How much money will you have on day 31? It will be more than enough to buy a new car: $21,474,836.47!

The calculation must take into consideration that the string of pennies began with 1-penny. Cumulatively on day 1 there was 1-penny; on day-2 there were 3-pennies; on day 3 the total was 7-pennies, etc.

So tap in the number 2 on your handy calculator; tap the YX key; then tap 31 followed by the = key. This will give you the number of pennies accumulated after 31 days.

But you started with 1-penny, so you have to subtract it from the total (the total has to be an odd number of pennies).

Once you arrive at the number of accumulated pennies, simply multiply by $0.01 to arrive at your HUGE payday amount!

Have a great week, and don’t spend this fortune wastefully; send some of it to me.

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