Video: Converting between Decimal Notation and Scientific Notation

A piece of dust has a mass of 0.0065 g. What is the mass of the piece of dust expressed in scientific notation to one decimal place?

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Video Transcript

A piece of dust has a mass of 0.0065 grams. What is the mass of the piece of dust expressed in scientific notation to one decimal place?

Okay, so in this question, we’ve been told the mass of a piece of dust, which is 0.0065 grams. And we’ve been asked to express this mass in scientific notation and to one decimal place. Because the way that it’s been currently written is been written to four decimal places and it’s in decimal notation. We can confirm this because there are one, two, three, four decimal places. And decimal notation is just basically when this number is written out in full.

Scientific notation, on the other hand, is when a number is written as 𝑎 times 10 to the power of 𝑏, where 𝑎 is a number that’s greater than or equal to one and is less than 10. And 𝑏 is an integer or a whole number. And so a number written in scientific notation consists of some number greater than or equal to one and less than 10 multiplied by 10 to the power of some integer. And we want to write the mass of the piece of dust in grams in the same way.

So let’s start with our original value, 0.0065 grams. Now because we want to write this exact number just in a different way, in scientific notation, we must keep this number exactly the same. And therefore, all we can do is to multiply or divide by one. And in this case, we’re going to be multiplying by one because 0.0065 times one is 0.0065. But then this is when the clever bit comes in.

One is the same thing as, say, for example, 10 divided by 10. And so if we take 0.0065 grams and multiply it by 10 divided by 10, then we’re still only multiplying our original number by one. And the reason that we do this is because we can then multiply 0.0065 by the numerator of our fraction, which is 10, and keep the denominator exactly as it is. And another way to write this is to write the 10 in the numerator in front of the 0.0065 grams. And so what we’re left with is 10 multiplied by 0.0065 grams multiplied by one divided by 10. But then 10 times 0.0065 gives us 0.065. And so what we’re left with is that our original number, 0.0065 grams, is equal to 0.065 grams multiplied by one over 10, at which point we can repeat our process again.

We multiply this whole thing by one or 10 divided by 10. Then we take the numerator of this fraction, stick it in front of our 0.065 so that we can multiply the 10 by 0.065. And this time, we can combine denominators. We can multiply 10 by 10 to give us a denominator of 100 or 10 squared. Doing all of this gives us 10 multiplied by 0.065 grams times one over 10 squared. And then 10 times 0.065 gives us 0.65. And so our original number, 0.0065 grams, is now equal to 0.65 grams times one over 10 squared.

And then guess what? We repeat the process once more, multiplying by 10 over 10. Combining the denominators, 10 squared times 10, and multiplying 0.65 by 10 gives us 6.5 grams multiplied by one over 10 cubed. At which point we’re going to stop multiplying by 10 over 10 because we’ve now found the value of 𝑎. It’s 6.5, because 6.5 is greater than or equal to one and less than 10.

So all that remains is to rearrange this bit so that we get something that looks like this, multiplied by 10 to the power of 𝑏. And of course, we can stick the unit of grams right at the end because we’re still dealing with some amount of mass, the mass of the piece of dust. And so writing everything in a slightly different way, we get 6.5 multiplied by one divided by 10 cubed grams.

And remember, because this whole thing is equal to 6.5 multiplied by one over 10 cubed multiplied by the unit of grams, we can switch the order of any of these around. Because multiplication is what is known as a commutative process, which is just a fancy way of saying that we can swap the order of two things that are being multiplied together. In other words, some number 𝑐 multiplied by some other number 𝑑 is the same thing as 𝑑 multiplied by 𝑐.

And so we can switch the order of one divided by 10 to the power of three and the unit grams, at which point everything that we’ve written down here is starting to look very close to being in scientific notation. All we need to do now is to recall that one divided by something to some power is equal to that thing to negative that power. In other words, one divided by 10 to the power of three is the same thing as 10 to the power of negative three.

And so we can write the mass of the piece of dust as 6.5 times 10 to the power of negative three grams. At which point the numerical value 6.5 times 10 to the power of negative three is in scientific notation. We’ve got 𝑎, which is a value that’s greater than or equal to one and less than 10, which happens to be 6.5 in this case. And we’ve got 𝑏, which is an integer. It’s negative three, which is a whole number.

At which point we can say that we’ve found our final answer, which is that the mass of the piece of dust is 6.5 times 10 to the power of negative three grams. Because this value is written in scientific notation and also, coincidentally, already happens to be written to one decimal place. And so that’s how we would go about converting from decimal notation, which in this case was 0.0065 grams, to scientific notation.

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