Scientific notation is a way of writing numbers in powers of 10 to express very large numbers and very small decimals. This is more convenient than writing these numbers out. It is oftentimes used in science, hence the name "scientific notation".
These numbers are written in the form of:
This is called exponential notation, because we have raised the number 10 to a power.
Let's say that we're analyzing a water sample and we find that it contains fluoride.
The amount of fluoride in the water is 4 parts fluoride per 1,000,000 parts water, or
Remember how to convert fractions to decimal form?
Whew! How tedious. You could also have found the decimal form by just moving the decimal point of the denominator, 1,000,000, to the left and moving the decimal point of the numerator, 4, to the left, until your denominator is equal to 1. This is hard for me to illustrate on the blog, so I am scanning in an illustration:
As you can see, I counted the number of times I had to move the decimal place of the denominator (6) and then moved the decimal point of the numerator by an equal number of decimal places (6).
So now we know that 4 parts per million is equal to 0.000004 in decimal form. That decimal is getting rather long however. What if we had 4 parts per billion? Counting out all those zeros could lead to error should you neglect just one zero. Scientific notation helps to avoid making these mistakes.
Instead, we can write out 4 parts per million as
To clarify, we can write 4 parts out of one million in:
Standard Notation: 0.000004
Exponential (Scientific) Notation:
Did you notice that the number of zeros in the standard notation of 4ppm is the same amount of 10's to the -6 expressed in exponential notation?
To raise 10 to the power of any number, you just need to write a 1 with the number of zeros that follow:
1 with 1 zero
1 with 3 zeros
1 with 5 zeros
is a bit of a special case. It basically means 1 with no zeros. For your reference, any number raised to the power of 0 is equal to 1, but we won't get into why until we start talking about exponents.
Anytime you see 10 being raised to the power of a negative number, you know that the result will be a decimal. Essentially we are saying that the number has a negative amount of powers of 10. Always remember to count the leading zero (the 0 to the left of the decimal point). Yet another reason why it is important to place a zero before any decimal point.
1 with 1 zero
1 with 3 zeros
1 with 5 zeros
Multiplying Powers of 10
To multiply powers of 10, we just find the sum of the exponents of the numbers by adding them and then rewrite them as a power of 10 with our new sum as the exponent.
Just as in regular addition, a number + 0 = original number
Dividing Powers of 10
To divide powers of 10, we find the sum of the exponents of the numbers by subtracting them and then rewrite them as a power of 10 with our new result as the exponent.
Remember that a double negative changes the sign of a number; our problem is -2 -(-6) = -2 + 6 = 4
Remember in the beginning when I said that scientific notation was written in the form of
?To have a number written in scientific notation, we usually have a coefficient in front of our power of ten.
How do we handle multiplying/dividing those coefficients, now that we know how to multiply/divide the powers of 10?
The answer is simple: multiply/divide the coefficients as normal.
You can see that we multiplied 4 x 2 as normal, and then added the exponents of the powers of 10.
Again, here, we divided 8 by 2 and then subtracted the exponents of the powers of 10.
That's about all there is to know about scientific notation. Remember that the exponents of the powers of 10 follow the same rules as any other exponent, which you will understand better when we get into exponents!
and 50% =
back to a percentage:
to a percentage:
to a percentage:
