feeling stupid, help me with powers of 10

Factual Questions
1

For some reason i can’t seem to remember to calcute powers of 10 correctly and it’s really killing me. I was looking at this flash animation about the size of the universe and i noticed the 10^-1 notation.

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/

The way i am thinking now is telling me that 10^-1 means moving the decimal point one place to the left. So 10.^-1 should become 1. right? No it’s not right and i know it isn’t, but why? 10^-1 is .1 and i’m not understand this.

2

10[sup]2[/sup] = 11010=100
10[sup]1[/sup] = 1*10 = 10
10[sup]0[/sup] = 1
10[sup]-1[/sup] = 1/10 = 0.1
10[sup]-2[/sup]=(1/10)/10 = 0.01

Simple as that. The exponent is the number of times you multiply ten together, or divide by ten in the case of a negative.

3

There’s also a fairly simply mode of figuring out what ten-to-the-x is written out as a number: Ten to any positive power is written, base ten, as one followed by the number of zeroes in the exponent; ten to any negative number is ten preceded by the number of zeroes in the exponent, with a decimal point following the first of them, keeping in mind that this system requires you to write a decimal as “0.nnn”

Similarly, 410[sup]4[/sup]=40,000 – four times one followed by four zeroes; 3.210[sup]4[/sup]=32,000 – three-point-two times one-followed-by-four-zeroes, which equals 32 times one-followed-by-three-zeroes.

4

Yeah…remember that all of the above is in relation to 1, since any number (except zero) to the zero power is one.*

So it’s true that you move the decimal point so many places to the left or right, but it’s how many places you move it from 1 that is important, not from 10.

  1. is 0 places from 1.

  2. is one place right of 1. (hence 10[sup]1[/sup])

.1 is one place left of 1. (hence 10[sup]-1[/sup])

  • When I think about why this is, I think about the fact that whatever number you think of is one times that number.

10[sup]1[/sup] = 1 times 10 once = 10
10[sup]2[/sup] = 1 times 10 times 10 = 100
10[sup]-1[/sup] = 1 divided by 10 = 0.1
and, so 10[sup]0[/sup] is 1 times nothing, or 1.

5

10[sup]-1[/sup] = .1 for the reason that 10[sup]m + n[/sup] = 10[sup]m.[/sup]10[sup]n[/sup]. So, 10[sup]0[/sup] = 10[sup]1 - 1[/sup] = 10[sup]1.[/sup]10[sup]-1[/sup]. Since 10[sup]0[/sup] =1, and 10[sup]1[/sup] = 10, then 10[sup]-1[/sup] must equal 1/10.

You can see that 10[sup]0[/sup] = 1, since 10[sup]1[/sup] = 10[sup]1 + 0[/sup] = 10[sup]1.[/sup]10[sup]0[/sup]. so, 10[sup]0[/sup] must equal 1.

6

Once you’ve got all that down, you can move into fractional powers. For example, what is 10[sup]0.5[/sup]?

Well, you can think of 0.5 as being 1/2, because that’s what it is: one divided by two. So, it’s 10 raised to the 1/2 power.

Now, what does that mean? Well, it means that 10 is being raised to the inverse of an integral power. So we do the opposite (or inverse) of the exponentiation function, which is taking a root.

Which root do we take? Well, look at it: 1/2. Since 2 is in the lower portion, we take the 2-th root, more commonly called the square root. Therefore, 10[sup]0.5[/sup] == sqrt(10), where sqrt(10) is as close to representing the square root function applied to 10 as I can come.

Now, the square root of 10 isn’t a `pretty’ number. It’s irrational, as a matter of fact, which means it goes on forever without repeating itself. It is, however, a surprisingly good approximation for the value of pi, which is an important number that’s also irrational. But none of that’s really important.

Now, what would 10[sup]0.75[/sup] be? Well, suss out 0.75: It’s 3/4. Now we have a variation. We have a number that isn’t one on top.

Work this out by first applying the familiar rule: The four is in the inverse position, so in this case we take the fourth root. But what do we do with that three? Since it’s in the upper position, we take it at face value: 10 gets raised to the third power. Therefore, 10[sup]0.75[/sup] is equal to the fourth root of 10[sup]3[/sup].

7

√10 (&****radic;10)

8

Derleth

Actually pi is a transcendental number (it cannot be the root of an ordinary algebraic expression) whereas square roots are irrational, as you correctly stated.

It may be nitpicking, but it was a story that had to be told.

9

wolf_meister: I thought pi was irrational and transcendental.

This looks like a proof of pi’s irrationality. – Reading the image is obnoxious. Download the PostScript or the PDF.

10

Bah, I should tie up loose ends:
[ul]
[li]Transcendental: Not the root of an ordinary algebraic expression.[/li][li]Irrational: Cannot be represented as the quotient of two integers.[/li][li]Rational: Can be represented as the quotient of two integers.[/li][li]Integer: A number that can be constructed by repeatedly adding one to, or subtracting one from, zero.*[/li][/ul]

*That’s the Peano axiom. Computer scientists are big fans of playing Peano. :slight_smile:

11

Every transcendental number is irrational, but there are many irrational numbers which are not transcendental.

And 0[sup]0[/sup] is usually taken to be 1, because otherwise the binomial theorem can’t be used to expand (a + 0)[sup]1[/sup]. We wouldn’t want that.

12

thanks for clearing it up for me guys

13

I thohught that when the operation was staright forward exponention 0[sup]0[/sup] was usually taken as indeterminate?

14

You can do it that way, but there’s a good reason to take it as 1, so why not?