Defining Powers of Ten

Date: 09/22/2004 at 09:22:03
From: Christina
Subject: Definition of the Powers of 10

Please clarify the definition of powers of 10.  I have read that a 
power of 10 is a number that can be expressed as only the product of a
certain number of 10s.  But I also think that 1/10, 1/100, or 10 to 
the negative first power are powers of 10.  Yet, they don't seem to
fit with that definition.  Which is correct?

Date: 09/22/2004 at 10:17:37
From: Doctor Ian
Subject: Re: Definition of the Powers of 10

Hi Christina,

The definition depends on the context.  For example, if we're just
discussing something like place values in integers, then we would use
"powers of 10" to mean 10 raised to any non-negative integer exponent,
i.e., 10^0, 10^1, 10^2, and so on.

When we start talking about place values in decimal expansions, we
expand the definition to include negative integers: 

  ..., 10^-3, 10^-2, 10^-1, 10^0, 10^1, 10^2, 10^3, ...

In the absence of any particular context, it would be surprising for
someone to use "powers of 10" to include negative exponents.  But
"surprising" isn't the same as "incorrect."

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/22/2004 at 22:28:46
From: Christina
Subject: Definition of the Powers of 10

Thank you for your quick reply.  The context I was thinking about was 
doing conversions within the metric system.  Is it correct to say that 
you multiply by a power of 10 when you are converting smaller units to 
larger units as well as when you are converting larger units to 
smaller units?  I truly appreciate your thoughts!

Date: 09/23/2004 at 07:24:43
From: Doctor Ian
Subject: Re: Definition of the Powers of 10

Hi Christina,

In that context, I would say that you "multiply or divide by powers of
10," but I think in that context it would also be correct to just say
"multiply," i.e., to interpret 10^-3 as a "power of 10."

But in the end, it's not really a precise term, and you can be correct
but still be misunderstood by others.

After all, the square root of 10 is 10^(1/2).  So can we call that a
"power of 10"?  Normally we'd call it a "rational power."  So instead
of just using "powers of 10," it would probably be a good idea to
always use one of the following:

  non-negative integer powers of 10:  10^0, 10^1, 10^2, ...

  integer powers of 10: 10^0, 10^(+/-1), 10^(+/-2), ...

  rational powers of 10:  10^(a/b), where a and b are integers

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/23/2004 at 08:26:29
From: Christina
Subject: Thank you (Definition of the Powers of 10)

Thank you so much! Your answer is very thorough and helpful.

Christina