In this explainer, we will learn how to use scientific notation and unit prefixes to multiply and divide values of physical quantities by various powers of ten.
When quantifying a physical value, such as mass, charge, time, or velocity, smaller values are represented as decimals.
Here, the grain of sand is a mere 0.004 grams. This has a few zeroes in it but is still easy to write out. However, the study of physics often deals with much smaller measurements than grains of sand.
Various physical quantities of subatomic particles are referenced regularly, such as the charge of a proton, mass of a neutron, or orbital distance of an electron. These values are so small that representing them in decimal notation is simply not practical. For example, here is the charge of a proton:
Writing out the decimal form every time would take up a lot of paper and increase the likelihood of missing a zero or two, so when dealing with very small numbers such as this, scientific notation is used instead.
Scientific notation allows us to write very small numbers in a neat way. Here is the charge of a proton expressed in scientific notation:
The power of the 10 in scientific notation represents how many spaces the decimal point moves until it is just to the right of the first nonzero number. For smaller values, this power will be a negative and will represent how many places the decimal point has moved to the right. For example, consider this number:
We see that the decimal in 0.013 has to move 2 spaces to the right to make it to the first nonzero number, which is 1, as shown below.
Normally, the zeros in front of the decimal point are not written. They are only here to show how the decimal point has moved. Since the decimal moved 2 spaces, the power of the 10 in scientific notation is :
Let’s look at another example on how to convert between the two notations with the grain of sand. The grain of sand is 0.004 grams, which looks like
We have to move the decimal point 3 spaces to get it to the right of the first nonzero number, as shown below.
Thus, our power in scientific notation will be .
Our mass of a grain of sand in scientific notation is
Let’s look at another example.
Example 1: Mass of Dust in 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?
The correct power of the 10 in scientific notation will be how many places the decimal point moves to the right. Since the decimal moves until it is to the right of the 6, making it 6.5, we can just count how many places it moves, as shown below.
The decimal point has moved 3 spaces, meaning the correct power for scientific notation is . This makes the answer in scientific notation look like .
The answer is thus B.
Converting the other way works just the same but in reverse. When given a number in scientific notation with a negative power of 10, move the decimal point a number of spots to the left equal to the power.
Say that we are looking at a particularly small grain of sand, a thousand times smaller than the other one, as shown below.
The value of mass we are given is grams. The power is , meaning we have to move the decimal point 6 places to the left to convert it to decimal form. Every empty space we move the decimal point, we replace with a 0, as shown below.
In decimal form, the mass is thus
The number of zeros placed in front of the 4 is related to the power of 10 in scientific notation. Specifically, it is one smaller than the power. So, a power of means 5 zeros are placed in front between the nonzero number and the decimal place. Observe the table below.
Expressed as a formula, the way you can convert small numbers between scientific notation and decimal form looks like where is the first nonzero number and is the power of the scientific notation. This formula only works on small numbers (i.e., when the exponent is negative).
Let’s look at another example.
Example 2: Time Taken by a Bullet to Come to Rest in Scientific Notation
A bullet comes to rest in a time of s. What is the time taken for the bullet to come to rest, expressed in decimal form?
For this value of time, we see that the power of the 10 is . This means there will be 3 zeros in between the nonzero number and the decimal point as it is moves to the left. In the equation the nonzero number, , is 5. The power is . Thus, is 4. The number of zeros in between the decimal point and is thus 3:
Expressed in decimal form, the time is thus
Even with scientific notation, the time taken to say a small value can still be cumbersome. Let’s look at the atomic radius of hydrogen:
This would be said as “two point five times ten to the power of negative eleven metres”. There is a shorter way of expressing these small values, using unit prefixes:
This would be said as “twenty-five picometres”. You have already used unit prefixes for some units before (e.g., milli- in milliseconds or millimetres). Below is a table showing the prefixes for small units.
The unit symbol is what is put before the actual symbol of a unit. One milligram would be 1 mg, for example.
The scientific notation column denotes what the power would be for 1 unit of that size. So, 1 picometre (pm) would be written as metres. The power also tells us what to divide a number by in order to get it in those units. For example, when converting 0.003 m to micrometres.
Putting it in terms of micrometres requires us to divide by :
Since this is a division, it means we move the decimal place 6 places to the right:
Let’s look at an example.
Example 3: Nanowatt in Scientific Notation
Which of the following is equal to one nanowatt when multiplied by one watt?
Looking at the prefix table above, the prefix nano- denotes a value of one billionth of whatever unit it is attached to. Written in scientific notation, the numerical value of this is .
Multiplying by 1 watt thus gives a value of 1 nW:
The correct answer is then A.
It is possible to represent units any way we please, but the reason these prefixes were created was to allow us to quickly and clearly compare numbers. When using them, we often want to use prefixes that give us the shortest nonfractional number.
To see this, let’s say we have one ten thousandth of a second. In decimal form, this looks like
To put this in terms of any other unit prefix, you divide it by 1 times the number given in the unit prefix table. For milliseconds, we would divide it by , which means moving the decimal point three places to the right:
This is much easier to see than one thousandth of a second, but it is still below 1. Let’s put it in terms of microseconds instead, then. For microseconds, the division would instead be by , moving the decimal place six places to the right:
This is easier to make quick comparisons with and see than 0.01 ms or 0.00001 s.
Let’s look at some examples.
Example 4: Length Unit Conversions in Scientific Notation
What is the value of 1 centimetre in units of micrometres?
Normally, to put metres in units of micrometres, we would divide by , but we are given centimetres instead. So, let’s convert 1 centimetre into metres. There are 100 centimetres in one metre, so 1 centimetre in terms of metres is
We can then take the value in metres and divide it by to obtain our value in units of micrometres. This is the same as moving the decimal 6 places to the right:
Expressed as micrometres, 1 centimetre is thus 10 000 μm. So, the correct answer is C, μm.
Example 5: Area of a Rectangle with Unit Prefixes
A rectangle has side lengths of 0.02 m and 0.004 m. Calculate the area of the rectangle.
- 80 μm2
- 8 μm2
- 80 nm2
- 8 nm2
The area of a rectangle is the product of the side lengths. Expressed in square metres, the area is simply but we want this amount in terms of either square micrometres, μm2, or square nanometres, nm2. To put this value in terms of square nanometres, we would divide it by : which means moving the decimal point 9 places to the right, since this is a division. This gives which is none of the answers! Let’s try square micrometres instead, so we divide by , meaning we move the decimal place 6 places to the right:
The answer is thus A, 80 μm2.
Let’s summarize what we have learned in this explainer.
- The way to convert small numbers between scientific notation and decimal notation is where is the first nonzero number and is the power of ten of the scientific notation.
- Unit prefixes are used to quickly and clearly show differences between small values. These prefixes are as follows:
Unit Prefix Unit Symbol Scientific Notation Milli m Micro Nano n Pico p Femto f
- To put a unit in terms of another smaller unit, divide by its representation, where is the power of the scientific notation.