Foundations- Lesson 75 Exponential and Scientific Notation
Part 1- Lesson 119 Scientific Notation
Part 2- Lesson 107 Scientific Notation
Scientific Notation is a way to write very large or very small numbers in an
abbreviated fashion. When you express a number in scientific notation you’ll have
a combination of any number greater than or equal to one but less than ten
multiplied by a power of ten. Let’s look at a few examples twenty three thousand
we need to have a number greater than or equal to one but less than ten multiplied
by a power of ten. We know that when we multiply by a power of ten the decimal
moves. Since we do not see a decimal on this number because it is a whole number
we know the decimal belongs after the very last digit to the right. Let’s place the
decimal here and move it as many times as we need to in order to get a number
greater than or equal to one but less than ten if we move the decimal place one
space to the left we have two thousand three hundred this number is greater than
ten which won’t work for scientific notation. If we move our decimal again to the
left one place we now have two hundred thirty. This number is still larger than ten.
If we move the decimal again we have twenty three with one more move of the
decimal we will have two point three or two and three tenths. This number is
greater than one but less than ten. Now we need to count the number of times we
moved the decimal altogether. We moved the decimal a total of four times one two
three four.
Another way to look at this is to start at the beginning. We move the decimal one
two three four this means we will raise ten to the fourth power and multiply two
point three by this number. So twenty three thousand can be expressed in scientific
notation as two and three tenths times ten to the fourth power. So to check I can say
two point three times ten to the fourth and move the decimal four places to the right
one two three four I will need to fill in these empty spaces with zeros and we are left
with twenty three thousand which is what we started with.
The next example is two hundredths. In scientific notation we need a number
greater than or equal to one but less than ten multiplied by a power of ten. In this
example we need to move the decimal two spaces to the right in order to get a
number greater than or equal to one and less than ten. Because we needed to move
the decimal to the right our exponent or power will need to be negative. Therefore
two hundredths can be written in scientific notation as two times ten to the negative
two.
Let’s check two times ten to the negative two tells us we need to move our decimal
two places to the left because the exponent is negative. Since it begins to the right of
the two we’ll move it one two places to the left and fill in with zeros so we have two
hundredths. Because we have no whole number we can place a zero as a place
holder and now we have exactly what we started with.
Look at it going the other direction. Problems like this will be asking us to write
numbers in standard form when given the scientific notation. Four and six tenths
times ten to the fifth power. This number is written in scientific notation and we
need it in standard form. To do this we move the decimal the appropriate spaces to
the left or right if the exponent is negative we move our decimal to the left, if the
exponent is positive we move the decimal to the right. Because the exponent in this
problem is positive we will move our decimal to the right five spaces one two three
four five and we fill in the rest with zeros so we have four hundred sixty thousand.
Four and six tenths times ten to the fifth power is four hundred sixty thousand
written in standard form.
Here’s another example four and six tenths times ten to the negative fifth power.
We need to look at the exponent and it’s sign to determine if we move the decimal to
the left or the right. Because this exponent is negative we’re going to move our
decimal to the left five spaces. Four and six tenths we need to move the decimal five
spaces to the left one two three four five and fill in the rest with zeros. This number
would be read forty six millonths.