Scientific notation is an application of power notation.
We use it to write very large or very small numbers in a compact and consistent way.
What we do is we write one digit of our number, the first non-zero digit on the left, and
then we write a decimal point.
Then we write the rest of the digits of our number.
And then, we multiply the number by a suitable power of ten, so that what we end up with
is numerically the same as what we started with.
You may hear the words “mantissa” and “order” being applied to numbers in scientific
notation.
You probably don’t need to know these words, but if you’re interested, this is what they
mean.
Let’s try some examples.
Write sixty thousand in scientific notation.
Well, let’s write the first digit of our number, that’s the six, and then—well
in this case there aren’t any more digits.
Okay, now what power of ten can I multiply six by, to get back to sixty thousand?
Hmm, let’s work it out step by step.
We’ve currently got six.
If we multiply by ten, we get sixty—that’s ten to the power of one.
Ten to the power of two gives us six hundred; to the power of three is six thousand; and
finally four gets us what we want, sixty thousand.
Six, times ten to the power of four.
Let me show you another one.
Write seven hundred and twenty-three million in scientific notation.
So, write the first digit, seven, then a decimal point, then the rest of the digits, two three.
Okay, now what power of ten do I have to multiply this by to get back to where I started?
Seven point two three times what power of ten gives me seven hundred and twenty-three
million?
Well, here’s seven point two three.
Multiply by ten: seventy-two point three; by a hundred, seven hundred and twenty-three;
and keep increasing the power of ten until we get what we want.
Three, four, five, six, seven, almost there.
Eight.
Seven point two three times ten to the power of eight is seven hundred and twenty-three
million.
Actually there is a quicker way to get here.
The first non-zero digit in our number is the seven.
What’s the place value of this digit?
What’s the place value of the seven?
It’s seven hundred million, right?
The place value is hundred millions, which is ten to the power of eight.
See?
Let’s try this method on the next example.
Six hundred and five.
Well, start by writing one digit, then the decimal point, then the rest: six point zero
five.
Okay, now what was the place value of the six?
Hundreds, which is ten times ten, or ten to the power of two.
Six point zero five times ten to the power of two.
Another: Eighty-two thousand.
Eight point two, times what power of ten?
Well, what’s the place value of the first digit?
Ten thousand.
That’s ten to the power of four.
So it’s eight point two times ten to the power of four.
One more like this, a little bit trickier.
Twenty-four point eight.
Well, it’s two point four eight, times what power of ten?
Look at the first digit: two, in the tens column.
So the power of ten is one.
Now I know that ten to the power of one is just ten; you don’t actually have to write
the one.
But in scientific notation we usually do write the power of one, just to be consistent with
the normal format.
So much for large numbers.
But scientific notation applies equally well to very small numbers.
Let’s try it.
Write zero point zero-zero-zero, zero-zero-four, in scientific notation.
Hmm, well it’s going to be four, times ten to some power.
What’s the power going to be?
Well, what’s the place value of that four?
It’s tenths, hundredths, thousandths; ten-thousandths hundred-thousandths, millionths.
Which is going to be ten to the power of negative six.
But just to show you that this works, here’s how to do it step by step.
We’ve currently got four.
Here’s four times ten to the power of negative one, which is zero point four; negative two,
zero point zero four; negative three, negative four, negative five, and finally negative
six: this is the number we started with.
So zero point zero-zero-zero, zero-zero-four, in scientific notation, is four times ten
to the power of negative six.
Try another: Zero point zero-seven-eight, one-four-three.
Okay, write one digit and then the decimal point: seven point eight one four three.
Now, what’s the place value of the first digit, the seven?
It’s hundredths, which is ten to the power of negative two.
Seven point eight one four three, times ten to the power of negative two.
Zero point zero-zero-zero, three-zero-six.
One digit, then the decimal point: three point zero six.
What’s the place value of the first digit?
Ten-thousandths.
Three point zero six, times ten to the power of negative four.
Okay, another tricky one.
Seven point zero one.
Hmm, how will write this in scientific notation?
One digit then the decimal is seven point zero one… and the place value of the seven
is just one, which I could write as ten to the power of zero.
(Remember, anything to the power of zero is one.)
(Actually, in this case, most people would just leave the power of ten off, leaving it
as seven point zero one.)
Alright, so how do we convert a number in scientific notation back into an ordinary
decimal number?
Well that’s easy: just apply the power of ten.
Write eight point three, times ten to the power of four as a decimal number.
I’ll do this first one for you step by step, one power of ten at a time.
This is eighty-three, times ten to the power of three; or eight hundred and thirty, times
ten to the power of two; eight thousand three hundred, times ten to the power of one; or
finally eighty-three thousand, times ten to the power of zero.
Note that we could have got straight here by applying the power of ten as the place
value of the first digit of our number: the eight goes into the position of ten to the
power of four, which is the ten-thousands digit: eighty thousand.
Then add the rest of the digits: Eighty-three thousand.
Write two point zero four one, times ten to the power of six, as a decimal number.
Well, the two goes in the ten-to-the-power-of-six place, which is the millions: Two million.
Then write the other digits: Two million, forty-one thousand.
Six, times ten to the power of negative three.
Ah, this is a negative power, so my answer is going to be smaller than one.
Negative three corresponds to thousandths, so this is zero point zero zero six.
Write nine point one three, times ten to the power of negative five, as a decimal number.
Okay, ten to the power of negative five is hundred thousandths; write the nine in this
place—it’s the fifth digit after the decimal point.
Then write the other digits where they belong.
Zero point zero-zero-zero, zero-nine-one, three.
Last example, a tricky one: two point six zero five, times ten to the power of two.
Well, ten to the power of two is a hundred, so write the first digit as two hundred.
Then fill in the other digits, carefully adding a decimal point in the right place.
Two hundred and sixty point five.