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.
We often encounter extreme values in our study of the universe—for example, it has been estimated that there are around 10 000 000 000 000 000 000 individual living insects on Earth. In the scientific community, as we record, perform calculations with, and share our findings, a number with so many zeros in it (19 here) is bulky, slow to write, and easy to copy down incorrectly. The value above is expressed in decimal notation, which is based upon representing a quantity by assigning a value to each power of ten in the decimal system. Each power of ten is represented by a “place,” and any single place in a number is assigned a value between zero and nine.
Let us examine another value written in decimal notation, such as 1 234.56. This number has six different places that represent different positions in the decimal system (or powers of ten, which we will explore later). To calculate what each place value contributes to a number, we multiply the value assigned to the place (which must be a single digit, zero through nine) by the decimal value, or power of ten, that each place represents. This information is expressed in the table below.
It can now be clearly seen how these separate places and their values combine to create one single quantity. Adding their contributions, we have . This is the main idea of decimal notation: each place, with its assigned decimal value, contributes a certain quantity to the number as a whole.
Each place in decimal notation represents a different power of of ten. For example, recall that the tens place multiplies its assigned value by 10 and the hundreds place multiplies its assigned value by 100. Each place value—1 000, 100, 10, 1, 0.1, 0.01, and beyond—can be expressed as 10 raised to a certain power. For instance, , , and . The smaller decimal places can also be expressed in this way. Remember that any number raised to zero equals one, so we can express the ones place value as . Further, decimals can be expressed as ten raised to a negative number. For example, we know that and .
It might already be intuitive that any large power of ten, which begins with one as the leading digit and is followed by a certain number of zeros, can be expressed as ten raised to a positive integer, and thus any quantity can be written as a one followed by zeros, so long as is a positive integer. For example, ten raised to the fourth power expands into decimal notation as a one followed by four zeros: . Likewise, we know that one billion (1 000 000 000) is a one followed by nine zeros, so we can express it as .
Expressing a number with a power of ten like this is much more compact and efficient to work with than counting and writing out many zeros. For example, let us consider the two large numbers 10 000 000 000 000 and 100 000 000 000 000. At a glance, these values might (incorrectly) appear to be equivalent, and in order to compare their magnitudes, we would need to take the time to carefully count out and record how many zeros each value has. But if we see the same two numbers expressed as ten raised to a certain power, they are and , respectively, and in this form it is immediately clear that the two values are not equivalent, and there is no guess work or time-consuming counting required to understand and compare their magnitudes.
While decimal notation is very common and generally suitable for representing smaller, more manageable numbers like 13 or 1 989, often when we deal with very large numbers, like the number of insects on Earth or stars in a galaxy, decimal notation is not the best representation. Thankfully, we can use the concept of place values as powers of ten, as we explored above, to efficiently represent large values of physical quantities in scientific notation.
Scientific notation can represent any quantity as a value between one and ten that is multiplied by some power of ten. The value between one and ten results from simply moving the decimal point of the quantity to just behind its leading (nonzero) digit, and this change in the decimal point’s location is made up for by multiplying the new decimal value by a power of ten. Let us apply this concept to an example.
Example 1: Converting a Large Value from Decimal to Scientific Notation
There are 225 000 atoms in a dust cloud in deep space. What is the number of atoms in the dust cloud, expressed in scientific notation to two decimal places?
In order for a quantity to be expressed in scientific notation, it must take the form of a value ranging from one to ten that is multiplied by an integer power of ten. The decimal point must be just behind the first nonzero digit, so we need to move it from its original place, as illustrated below.
The decimal point has to move five places to the left to get just behind the first digit of our number. With the decimal place moved, the value reads 2.25000, or just 2.25.
As to not change the overall value of the quantity, we must represent all those decimal places that we jumped over with some power of ten. Since each decimal place represents a different power of ten, and we jumped over five places, we can represent this as . We now have a power of ten and a value between one and nine (2.25) to multiply together.
Thus, the number of atoms in the dust cloud can be expressed in scientific notation as .
To check our work, we can expand this quantity back out into decimal notation, using the fact that :
This example demonstrated how we can move the decimal point in a number, so long as the shift in the decimal point’s location is made up for by multiplying by the correct integer power of ten. Let us formally define the style of this notation.
Definition: Scientific Notation
A number is expressed in scientific notation if it is in the form , such that is a number greater than or equal to one and less than 10 and is an integer.
Scientific notation is commonly used in the scientific community to express very large or very small values. It is important to recognize quantities written in this notation, and it is also helpful to know how to convert from scientific notation into decimal notation. This process applies the same concepts we have learned thus far—we must simply recognize which way to move the decimal point and remember to move it the number of places that ten is raised to. Therefore, if we are given a quantity in scientific notation that has ten raised to a positive value, we know that, in order to write the quantity in decimal notation, we move the decimal point to the right, creating a larger decimal value and eliminating the need for a “power of ten” term.
Example 2: Converting a Large Value from Scientific to Decimal Notation
A block of ice is heated with J of energy. What is the number of joules of energy supplied, expressed in scientific notation?
This quantity of energy is expressed as a value between one and ten that is multiplied by an integer power of ten, so we know it is currently written in scientific notation. During our conversion, we will focus on the number associated with the quantity, but we must not forget that it also has units. The decimal point is currently located behind the first digit of the number (three here), and the term tells us that it must be shifted six places to the right in order to reach decimal notation.
The six jumps of decimal places, or powers of ten, have been counted out, so we will now place the decimal point in its new position, and we have reached decimal notation.
Thus, the value J can be expressed in decimal notation as 3 250 000 J.
When working through the example above, to avoid confusion as to which direction to move the decimal point, recall that . This means the original decimal value, 3.25, is multiplied by a large number, which is why the decimal point needed to be moved to the right, rather than to the left. This is worth restating to remember when converting between decimal and scientific notation.
Moving a decimal place to the right creates a decimal value of larger magnitude. Moving a decimal point to the left creates a decimal value of smaller magnitude.
Example 3: Converting a Quantity to Decimal Notation
A road is m long. Which of the following is equal to the length of the road?
- 75 m
- 750 m
- 7 500 m
- 75 000 m
- 750 000 m
The notation of might look familiar, but the quantity is not technically expressed in scientific notation since the value 0.75 is not between one and ten. Nevertheless, the method of multiplying a decimal value by powers of ten is still the same, and so we can expand this value into decimal form in the same way as we would do to a number in scientific notation.
We will start with the decimal value 0.75 and count four decimal place jumps to the right because we are multiplying our decimal value by , which is a large number. Recall that moving a decimal point to the right results in a larger decimal value, whereas moving it to the left creates a smaller decimal value, which would be incorrect here. This step is shown below.
We have counted out four decimal place jumps to the right, so we can place the decimal point in its new location. Thus, we have our value expressed in decimal form, as shown below.
Therefore, a length of m can be equivalently expressed as 7 500 m.
Now that we have practiced expressing scientific notation and powers of ten quantitatively, let us consider another way we can imply the same concepts using language. To begin, refer to the image below where a distance along a stretch of road is conveyed in kilometres (km).
The road sign could just as well use another unit of length that we are familiar with, like centimetres or plain metres, but it is much more convenient to write and read “2 km” than “200 000 cm” or “2 000 m.” All of these measurements are technically valid because they convey equivalent quantities; however, some make more sense to use than others because it is generally easier to keep track of and perform calculations with numbers that have more compact decimal values. This concept should sound familiar: we can rewrite a large quantity to have a more manageable decimal value, so long as we make up for this change by modifying the base measurement.
Grouping a unit by a power of ten can be expressed with a unit “prefix,” which is the part of the word that comes before and modifies the unit. In this instance, “metres” is our base unit, and we have seen the prefixes “kilo-” and “centi-,” which correspond to grouping by and respectively. Different powers of ten are represented by specific prefixes and symbols, as listed in the tables below.
Which prefix we use is up to us, and it is generally helpful to choose a prefix that will mimic scientific notation so that a quantity’s resulting decimal value is between one and ten. The table on the left shows prefixes used to describe very small quantities, like “nano,” which groups a quantity by billionths, or , but for now we will consider large values of physical quantities.
To use a prefix in a measurement, we must divide our amount of the base unit quantity by the corresponding power of ten and attach the prefix symbol to the unit abbreviation. For example, if we want to write 4 500 000 W using the prefix “mega,” we would need to divide the quantity 4 500 000 by and attach the symbol M to the unit abbreviation. Thus, we know , which is quicker and easier to write and record.
Notice that we have only used prefixes alongside units. Prefixes cannot stand alone in modifying a number—if we say we have a “milli” of something, it is not at all clear what the quantity is, and so we must include the unit of whatever we are measuring so that we have a millimetre or a milligram, for example.
Let us practice this concept with a few examples.
Example 4: Expressing Powers of Ten Using a Prefix
Which of the following is the number of hertz in a gigahertz (GHz)?
The prefix “giga,” symbolized by G, implies that we are multiplying our units, hertz, by , or a one followed by nine zeros. Therefore, we know that there are hertz in a gigahertz.
Example 5: Expressing Powers of Ten Using a Prefix
Which of the following is the number of watts in a terawatt (TW)?
The prefix “tera,” symbolized by T, implies that we are multiplying our units, watts, by , or a one followed by twelve zeros. Therefore, we know that there are watts in a terawatt.
Example 6: Expressing Powers of Ten Using a Prefix
Which of the following is the number of picojoules (pJ) in a joule?
The prefix “pico,” symbolized by p, implies that we are multiplying our units, joules, by . Therefore, one picojoule is equal to of a joule. This relationship can be expressed mathematically:
We want to know how many picojoules are in one joule, so we can divide both sides of the equation by to solve for 1 J:
This can be rewritten as
Thus, there are picojoules in one joule.
To finish, let us summarize a few important concepts.
- Scientific notation is useful for representing extreme values, as opposed to decimal notation, which assigns a value to each decimal place in a quantity, often including many zeros and making it inefficient to work with.
- A number is expressed in scientific notation if it is in the form , such that is a number greater than or equal to one and less than 10, and is an integer. To convert a large quantity from decimal to scientific notation, we move the decimal point to just behind the first nonzero digit, and a resulting decimal value is expressed as . We then represent the change in decimal point location by ten raised to the power of the number of places the decimal point moved, .
- We can use a prefix to imply that we are grouping a unit by a certain power of ten, which can help make an extreme quantity more manageable. To use a prefix, we divide the base unit quantity by the prefix’s associated power of ten and attach the prefix symbol to the unit abbreviation.