L3:__ Scientific Notation__

-Convention of Expressing Any
Base 10 Number As a Product of a Number Between One and 9, multiplied by 10 to
the Power of Some Exponent.

__Exponentiation__ is the replication of multiplication the way
multiplication is the replication of addition

10^{0} = 1

10^{1} = 10 10^{-1}
= 1/10 = 0.1

10^{2} = 10(10) = 100 10^{-2} = 1/[10(10)]=1/100=0.01

10^{3}=10(10)(10) = 1000 10^{-3}
= 1/[101(10)(10)] = 1/1000 = 0.001

__Advantages of Scientific
Notation: __

1. Allows Awkwardly
Large and Small Numbers to Be Expressed in Term
of Compact and Easily Written Numbers

2. Allows Accurate
Representation of the
Number of Significant Figures in a Number, That Is a
Measurement’s Precision, the “Certainty”
of Our Measurements

Convert the following numbers
to scientific notation

0.00456 and 456.00

A) 0.00456 Step
one, since this number is less than 1 you need identify the power of 10 which
you can multiply it by and set one digit to the left of the decimal. Here, we can multiply by 1000 or 10^{3 }and
force the original number to have one digit to the left of the decimal, but we
have changed its value.

0.0045610^{3} = 4.56 =(which is not equal to the original number)

Since we multiplied it by 10^{3}
we must also divide by 10^{3} (effectively multiplying the original
number by 1 and not changing it's value). Than we express the factor in the denominator
as a power of 10.

_{}

Note, once you get the hang
of this you can just count the number of digits you need to move the decimal to
give the original number one digit to the left of the decimal and multiply that
number by 10 to the negative value of those digits. Here we moved the decimal three positions to
the right, so we multiply by 10^{-3}.

B. 456.00, Since this number
has a value greater than 9 we must divide it by a factor of 10 to make the
first non-zero digit value between 1 and 9.

_{}

Note, once you get the hang
of this you can just count the number of places you move the decimal to the
right (giving a value with one digit to the left of the decimal) and multiply
by 10 to the power of that number (2 in this case).

__Using Scientific Notation
to Express Significant Digits__:

How can you express the
number 400 to 2 significant digits?

You must use scientific
notation.

4.0 x 10^{2}

As scientific notation always
has a decimal point there is never a problem expressing significant digits.

__Multiplication of numbers
in Scientific notation__

Try plugging the following
number into your calculator. Odds are
you will get an error message.

_{}

You will often get numbers
which are bigger or smaller than a calculator can handle. You need to treat the power separately from
the number

_{}=

__Addition and Subtraction
of Numbers in Scientific Notation__

Consider the following
problem

4.860 x 10^{12} + 9.7 x 10^{10}
+ 3.68x10^{11}

You first need to express all
numbers to the same power so you can line up the decimal point. It is suggested that you choose the largest
power and make everything else a fraction.

_{}