Associated Topics || Dr. Math Home || Search Dr. Math

### Scientific Notation

```
Date: 09/16/97 at 00:42:42
From: Aubin
Subject: Scientific notation

How do you do this problem? I don't understand how you're supposed to

5x10 to the 3rd power = 5,000
5x10 to the -3rd power = -5,000

Is this correct?
```

```
Date: 11/03/97 at 09:56:46
From: Doctor Pipe
Subject: Re: Scientific notation

Aubin,

The first part of what you wrote is correct; 5x10 to the 3rd power =
5,000. The second part is not correct.

Writing a negative exponent, such as 10^-3 (read that as ten to the
minus third power) is the same as writing 1/(10^3) (read that as one
over ten to the third power). Notice that the exponent is negative
when writing 10^-3 and positive when writing 1/(10^3) - yet the two
numbers are equal.

Remember that any number to the zeroeth power, say 10^0, is equal
to 1.  10^0 = 1; 5^0 = 1; 275^0 = 1.

Remember also that when multiplying two numbers written as
base^exponent, if the base in both numbers is equal then we add
together the exponents: 10^5 x 10^6
= 10^(5+6)
= 10^11.

If we have a number 10^5, what number do we multiply it by to get 1?
Well, 10^5 x 10^(-5)
= 10^(5 + (-5))
= 10^0
= 1.

So if 10^5 x 10^(-5) = 1
then         10^(-5) = 1 / 10^5

So, since 10^3 = 1,000 then 10^(-3) = 1/(10^3) = 1/1,000 = 0.001 .
It follows from this that:

5x10 to the -3rd power = 5 x 10^(-3) = 5 x 0.001 = 0.005 .

The reason for this can be seen by examining what numbers to the right
of the decimal point represent. You know what numbers to the left of
the decimal point represent: the units digit represents the numeral
times 10^0 (any number to the 0th power is 1), the tens digit
represents the numeral times 10^1, the hundreds digit represents the
numeral times 10^2, and so on.

Well, to the right of the decimal point, the tenths digit represents
the numeral times 10^-1, the hundredths digit represents the numeral
times 10^-2, the thousandths digit represents the numeral times 10^-3,
and so on.

It's important to understand exponents because exponents allow us to
extend the range of numbers that we can work with by allowing us to
easily write and work with very large and very small numbers.  It's so
much easier to write:

10^23

then to write:

100,000,000,000,000,000,000,000

Or to write:

10^(-23)

0.00000000000000000000001

Good luck and enjoy!

-Doctor Pipe,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Large Numbers
Elementary Square Roots
Middle School Exponents