Hi …Rob here with Math Antics.
…just wanted to say “hi” to all the people who watch our videos,
and say thank you so much for watching and liking and subscribing.
I know we usually don’t do milestone videos,
but this seemed like a pretty cool milestone that we just had to mention it.
We have one times ten to the six YouTube subscribers!
That’s crazy! That’s awesome!
Thank you SO much!
Oh wait…
You don’t know what numbers like one times ten to the six even mean?
Well you’re in luck!
We have a video that explains it.
In fact, it’s this video that you’re watching right now!
Hi I’m Rob. Welcome to Math Antics.
Have you ever heard people use numbers like that?…
1 times 10 to the 6th or maybe 3.4 times 10 to the negative 8th?
Well… those are examples of a way of writing numbers called Scientific Notation.
Huh… that didn’t sound quite right, let me try that again.
…Scientific Notation…
Ah yes… that’s better.
Numbers can be really big or really small, right?
Like, if you wanted to count up all the cells that make up your body,
it would be a really big number.
…something like 35 trillion cells!
But if you wanted to measure the diameter of one of those cells using meters,
you’d get a really small number.
…something like 0.0000005 meters.
Not only are really big or small numbers a lot of work to write down because of all the zeros,
they’re hard to quickly evaluate and compare.
At a glance, it’s not easy to tell just how many number places there are
in these really big or small numbers.
And that’s where Scientific Notation can really help us out!
Instead of using a long sequence of decimal digits to represent numbers,
Scientific Notation uses a shorter number multiplied by a power of 10.
And it’s always in that form: some number “…times 10 to the…” some exponent.
Here’s an example of a really long number: One-hundred, twenty-five million.
And here’s the equivalent number written in Scientific Notation: 1.25 times 10 to the 8th
Wanna see how these two numbers are just different ways of writing the same thing?
Let’s start by making a copy of our big number and messing with its decimal point a bit.
Where’s the decimal point you ask?
Remember that it’s always here, immediately to the right of the ones-place.
We just don’t need to show it if there aren’t any decimal digits.
Okay, so what would happen if we shift the decimal point one place to the left?
Well, doing that would change the number, right?
By definition, the decimal point is always immediately to the right of the ones-place,
so shifting the decimal point shifts the ones-place and all the other number places too.
And if we line up the ones-place of our new number with the ones-place of the original number,
you see that the new number is 10 times smaller.
That means shifting the decimal point one place to the left is equivalent to dividing a number by 10.
But, do we want a number that’s 10 times smaller than before?
Well…no. We don’t want to change the value of the number at all.
We just want to write it in a different way.
Since shifting the decimal resulted in a number that’s 10 times smaller than before,
to compensate and keep the value the same, we need to multiply the new number by 10.
Making the number smaller and then compensating for that might seem like a weird thing to do,
but it will make more sense in a minute.
Let’s do that process again.
Let’s make a copy of the new number and shift the decimal point to the left again.
Since that shifts all the number places, we can align the ones-places and see that the same thing happened.
The new number is 10 times smaller than before.
So to keep it the same value as the original number, we need to compensate by making it 10 times bigger.
We need to multiply by another 10.
And if we repeat that process again…
if we make another copy and shift the decimal point again,
we’ll see that the number gets 10 times smaller,
so we need to compensate by multiplying by another 10.
Alright, time out! We seem to have a little problem here.
Each time we shift the decimal point to the left, our number gets smaller.
But since we have to compensate with a factor of 10 each time, it’s making kind of a mess!
This part is getting shorter, but this part is getting longer!
No problem… Exponents can fix that!
Do you remember that exponents are a way of writing repeated multiplication?
If you don’t, then be sure to watch our videos about them before moving on.
Instead of writing 10 times 10, we can write 10 to the 2nd power,
and instead of writing 10 x 10 x 10, we can write 10 to the 3rd power.
That’s much better! Now we can continue on.
We’ll shift the decimal point again and multiplying by 10 again.
But this time, instead of writing another “times 10”,
we can just increase the exponent by 1 since there would be
a total of four 10s being multiplied together.
Let’s keep going with this process of shifting the decimal point to the left
and multiplying by 10 for each number place we shift.
And we’ll stop when there’s only one digit remaining to the left of the decimal point.
Wow! That’s quite a pattern.
Each time we shifted the decimal, the number got 10 times smaller.
So, each time we had to multiply by another 10 to keep the value the same.
And because we did that, each one of these lines represents the same value.
So even though it looks a lot different, this last line has the exact same value as the first one.
In fact, it’s just the original number written in Scientific Notation.
But, why is it only this last line, and not any of the others?
I mean… they all look pretty scientific to me!
Ah… that’s a good question.
For a number to be in ‘proper’ Scientific Notation,
it’s supposed to have only 1 digit to the left of the decimal.
There can be more than one digit to the right of the decimal, depending on the accuracy of the number,
but just 1 digit to the left.
But why? I mean… that rule sounds kinda arbitrary.
It’s not arbitrary at all!
If there's more than one digit to the left of the decimal point,
that would mean that we didn’t get out all of the factors of 10 that we could have.
And factoring out ALL of the 10s helps us quickly determine a number’s “order of magnitude”.
Order of magnitude!?
What in the world is that? That sounds kinda scary!
Order of magnitude is basically just how many 10s you need to multiply to get a certain number.
And when a number is in Scientific Notation, the order of magnitude is just the exponent,
because that’s telling us how many 10s to multiply together.
In this example, the Scientific Notation says that if we take this small decimal
and multiply it by eight 10s, we’ll get our original number.
So Scientific Notation is a way of taking a really big number
and reducing it down to a value that’s less than 10,
but keeping track of how many 10s we would need to multiply together to get the full number.
You can think of it as basically just extracting its “order of magnitude” and storing it in exponent form.
But why would we want to do that? I mean… it seems kinda complicated.
Well… yeah, but… did you see how much writing it saved us?
When we wrote out 125,000,000 we had to write ELEVEN characters, including commas.
But when we wrote the same number in Scientific Notation, we only had to write EIGHT characters.
What? …not convinced that it’s worth the savings?
Well how about this number?…
That’s a LOT of zeros to write, isn’t it?!
But in Scientific Notation, this number is just 8.4 times 10 to the 31st power.
That’s much better!
So Scientific Notation is very useful when it comes to writing down really large numbers…
or really small ones.
For example, this number is really small: 0.00000095
It’s much less than 1, but it’s not zero.
And here’s the same number written in Scientific Notation.
Again, it consists of a number that has only one digit to the left of the decimal point
which is being multiplied by 10 to a certain power.
But do you notice anything different about the exponent?
Yep… it’s negative! So what does that mean?
Well, the short answer is that positive exponents show repeated multiplication
while negative exponents show repeated division.
And since this is a negative exponent with 10 as the base,
it means to repeatedly divide by 10.
To see how that works,
let’s copy our original number and do that decimal-point-shift-thing again.
Only this time, we’re gonna shift the decimal point to the right.
What happens if we shift it one place to the right?
It makes the number 10 times bigger than it was before.
There used to be 6 zeros between the decimal point and the '9', but now there’s only 5.
Again, we don’t want to change the value,
so what can we do to compensate for shifting the decimal point?
In this case, since shifting one place to the right made the number 10 times bigger,
we need to compensate by dividing the number by 10.
And because of the way multiplication and division are related,
dividing by 10 is the same as multiplying by 1 over 10 (or one-tenth)
so we can just multiply by one-tenth.
OR… we can multiply by 10 to the negative 1,
because 10 to the negative1 is just another way of writing one-tenth.
That may seem odd if you haven’t learned about negative exponents before,
and we explain it in more detail in our video about the Laws of Exponents.
For now, all you really need to know is that
multiplying by 10 to the negative 1 is the same as dividing by 10,
so it compensated for shifting the decimal point one place to the right.
Continuing on, if we shift the decimal point another place to the right, the same thing happens…
we make the number 10 times bigger.
So to keep the value the same, we have to multiply by another factor of 10 to the negative 1.
And it you’re wondering whether we can combine these exponents, you’re on the right track!
10 to the negative 1 times 10 to the negative 1 combine to become 10 to the negative 2,
which makes sense because we shifted the decimal point a total of two places to the right.
And if we shift the decimal point 3 places to the right,
we need to multiply by 10 to the negative 3 to compensate.
And if we shift 4 places, then we need 10 to the negative 4 to compensate. …get the idea?
And if we continue doing that until the decimal point is
positioned so that there’s only 1 digit to the left of it,
that gives us the number in Scientific Notation: 9.5 times 10 to the -7
And can you figure out what the order of magnitude of this number is?
Yep… just like before, the exponent tells us.
It’s negative 7.
Being able to quickly identify a number’s order of magnitude is pretty handy.
For example, if the order of magnitude is a big positive exponent,
then you know right away that you’re dealing with a really big number.
But if it’s a big NEGATIVE exponent,
then you know you’re dealing with a really small number.
And if you’re comparing two really big number like these two,
or two really small numbers like these two,
it’s hard to tell at a glance which is actually bigger or smaller.
But if you see them in Scientific Notation,
it’s easy to see that this number’s order of magnitude is bigger than the other’s,
which means that it’s bigger,
and this number’s order of magnitude is less that the other’s
which means that it’s smaller.
So now that you’ve seen how Scientific Notation works,
and you realize that it’s just a short-hand way of writing really big or really small numbers,
let’s break down the procedure for converting back and forth
between numbers written in regular form and Scientific Notation.
Starting with these two examples in regular form…
First, count how many number places you would need to shift the decimal point
for there to be only 1 digit to the left of it.
For this number we’d need to shift 8 places,
and for this number; 6 places.
The number of places you need to shift will be the exponent in the Scientific Notation form,
but the sign of that exponent is determined by the direction you shifted.
If you shifted to the left (because you started with a big number) then the exponent will be positive.
But if you shifted to the right (because you started with a small number) then the exponent will be negative.
So that gives you the “times 10 to the something” part of the Scientific Notation
and to get the number that’s multiplied by that power of 10,
you just take the shifted decimal number and remove any zeros that don’t really need to be shown.
There… that wasn’t too hard, was it?
But what if we start out with numbers in Scientific Notation and want to convert them into regular form?
Let’s do that with these two examples.
The first step is to look at the exponent which is the order of magnitude of the number.
It tells you how many 10s you'll need to multiply or divide by to get the number in regular form.
If the exponent is positive, it means that you’ll need to multiply by that many 10s.
In this example, that means that we would need to
multiply by a total of seven 10s to get the number in regular form.
That would be a lot of work, but we can also just shift the decimal point that number of places.
Which direction do we need to shift it?
Well… since we’re multiplying by factors of 10,
we need to shift it in the direction that will make the number bigger.
That is, we need to shift it to the right.
So, we’ll just shift the decimal point 7 places to the right.
But as you can see, there’s aren’t 7 digits after the decimal,
so any places that don’t have a digit will just be filled with a zero.
There… Our number is regular form is 41,650,000
In this second example, the exponent is negative,
which means that we’ll need to divide by that number of 10s to get the number in regular form.
Again, we could just do that division, or we can shift the decimal point to save time.
Since dividing by factors of 10 make a number smaller,
we’ll need to shift the decimal point in the direction that results in a smaller number… that is, to the left.
Since our exponent is negative 5, we’ll shift the decimal point 5 places to the left.
And any places that we shift past that don’t already have a digit in them will get filled with a zero.
We’ll also put a zero in the ones-place since that’s always good form for decimal numbers.
There… Our number in regular form is 0.0000109
Alright… that’s the basics of Scientific Notation.
It’s may seem a little confusing at first, but as you get more experience with it, it makes a lot of sense.
And when it comes to writing really big or really small numbers, it’s totally worth it.
And even if you understand how Scientific Notation works,
it may take some practice to get good at converting back and forth between it and regular form,
so be sure to practice on your own.
As always, thanks for watching Math Antics, and I’ll see ya next time.
Learn more at www.mathantics.com