Video Transcript
The diameter of a particle is 0.02 millimeters. What is the diameter in nanometers? Give your answer in scientific notation.
The diameter is essentially the width of something, passing through the center of the body. Often, we use meters to measure objects, but if the object in question is very large or very small, then this is not always convenient. This table shows some helpful conversions. A millimeter is 1000 times smaller than a meter. Likewise, a micrometer is 1000 times smaller than a millimeter. This means that micrometer is a million times smaller than a meter. Furthermore, a nanometer is 1000 times smaller than a micrometer. This means that a nanometer is a billion times smaller than a meter.
The question gives us the diameter in millimeters and asks for the diameter in nanometers. To convert from millimeters to nanometers, we would need to divide by 1000 twice or divide by a million once. So a nanometer is a million times smaller than a millimeter. Therefore, one millimeter is equivalent to one million nanometers. We can use this conversion to convert the diameter in millimeters to nanometers. The question tells us that the diameter is 0.02 millimeters. We need to multiply this by a million nanometers divided by one millimeter. If we perform this calculation, the millimeter units cancel and we get a value of 20000 nanometers.
So we’ve now calculated the diameter in nanometers. But the question asks for a scientific notation. Thus, we need to convert this value to standard form. It needs to be written in the form 𝑎 times 10 to the power of 𝑛, where 𝑎 is greater than or equal to one and less than 10 and 𝑛 is a positive or negative integer. We can break apart numbers into different powers of 10. For example, 100 is equivalent to 10 times 10 or 10 to the power of two. And 1000 is equivalent to 10 times 10 times 10 or 10 to the power of three. If we had 2000, for example, then we could write it as two times 1000. And as we’ve already seen, 1000 is equivalent to 10 to the power of three. So we could also write 2000 as two times 10 to the power of three.
Two times 10 to the power of three is in scientific notation or standard form where 𝑎, which must be greater than or equal to one and less than 10, is two and 𝑛, which must be a positive or negative integer, is three. Let’s now try to write 20000 in standard form. 20000 is equivalent to two times 10000. 10000 as a power of 10 is 10 to the power of four. So 20000 in standard form is two times 10 to the power of four. This means that 20000 nanometers must be two times 10 to the power of four nanometers.
There is another way to convert to scientific notation or standard form. If we start with the 20000 nanometers and add in the decimal point, we can imagine the decimal point moving. We need to count how many decimal places this decimal point would need to move to convert 20000 into an appropriate value for 𝑎, which is greater than or equal to one and less than 10. If the decimal point was here, the number would be two, which is an appropriate value for 𝑎. If we count how many places the decimal point would have to move, we can see that it’s four. This value becomes 𝑛, which is what we found previously. We can now be sure that our answer has been correctly written in scientific notation.
So the answer to the question “What is 0.02 millimeters in nanometers in scientific notation?” is two times 10 to the power of four nanometers.
