1.2: Scientific Notation

Sometimes we need express very large or very small numbers. For instance, how many cells are there in the human body? And how large, in inches, is a cell? The answers to questions like this can be difficult to understand when expressed in regular notation:

Number of cells in the human body \( = 37,200,000,000,000\)

Length of a cell \( = 0.0011811\) inches

For numbers like this, it can be helpful to express them in scientific notation.

In this section, you will learn to:

• Recognize and explain scientific notation
• Use scientific notation to compare numbers

Basics of Scientific Notation

The goal of scientific notation is to make numbers shorter to write and easier to compare. It uses two tools you saw in the previous chapter: the base \(10\) system, and exponents. Let's start with the definition and then see how it is used in a few examples.

Definition: Scientific Notation

A number is written in scientific notation when it is expressed as a number between \(1\) and \(10\) that is multiplied by a power of 10. As an example, we could write

\[37,200,000,000,000 = 3.72 \times 10^{13}\]

The exponent on the 10 tells you how many places the decimal point has been moved. In this case, the exponent was moved 13 places to the left.

Scientific notation leverages the base \(10\) system to write very large or very small numbers in a compact way. In the definition above, we see that a positive exponent moves the decimal point to the left. Think for a moment about why this makes sense, based on the last section. Then try the following example.

Example \(\PageIndex{1}\)

Explain why the numbers \(37,200,000,000,000\)  and \(3.72 \times 10^{13}\) are equal. 

Solution

Remember from the previous section that exponents represent repeated multiplication, and we can tell a power of \(10\) by the number of zeroes has following the starting \(1\). Therefore,

\[10^{13} = 10,000,000,000,000 \]

When we multiply this number by \(3.72\), we can see (using either long-hand multiplication, or by thinking about the base 10 expansion), that

\[3.72 \times 10,000,000,000,000 = 37,200,000,000,000\]

Therefore, \(37,200,000,000,000 = 3.72 \times 10^{13}\).

The ideas above will allow us to write any number in scientific notation. The only difference for very small numbers -- those expressed as long decimals -- is that we use negative exponents instead of positive exponents. The following table summarizes how to write both very large and very small numbers in scientific notation.

Number Size	Exponent	Decimal Point Moves...	Example Number	Scientific Notation
Very Large	Positive	Left	\(453,000,000\)	\(4.53 \times 10^8\)
Very Small	Negative	Right	\(.000000674\)	\(6.74 \times 10^{-7}\)

Let's practice with one more example.

Example \(\PageIndex{2}\)

Convert the number \(.0000049\) to scientific notation.

Solution

We see that this is a very small number, between \(0\) and \(1\). This means that we need to use a negative exponent to move the decimal point to the right. We need to move the decimal point from its current location to the place between the \(4\) and the \(9\), so that the resulting number will be between \(1\) and \(10\). If we count places from the current location to the new location, we see that it has to move 6 places. Therefore, \[.0000049 = 4. \times 10^{-6} \]

The process of converting from scientific notation to ordinary notation consists of simply reversing the direction that the decimal point travels, and including an appropriate number of zeroes. Let's see two examples of this. Before looking at the solution, see if you can work backwards to figure it out yourself!

Example \(\PageIndex{3}\)

Convert each of the following to ordinary notation.

1. \(9.362 \times 10^{10}\)
2. \(5.7 \times 10^{-4}\)

Solution

1. There is a positive exponent here, which means we expect a very large number to result. We will move the decimal point 10 places to the right, since we are reversing the process of conversion. We will start putting zeroes once we run out of numbers in \(9.362\). Therefore, \[9.362 \times 10^{10} = 93,620,000,000\]
2. Here, we have a negative exponent, which means that our result will be a very small number. Since we are reversing the process, we'll move the decimal point 4 places to the left, adding zeroes as we go. Therefore\[5.7 \times 10^{-4} = .00057\]

What do "very large" and "very small" mean? As is often the case with topics in this book, the answer depends on the context in which you are using the number. The choice to write \(453,000,000\)  versus \(4.53 \times 10^8\) is up to whomever is writing that number. In general, they should consider the purpose of the number they are writing and who is likely to read it. In the context of a news article, one might see \(453,000,000\), or perhaps "453 million." The second version of the number gives a sort of alternative scientific notation, in which we've written \(453,000,000 = 453 \times 10^6 = 453 \times 1,000,000\). The previous equation does not give the standard scientific notation — which requires that the number in front be between \(1\) and \(10\) — but it can be a useful way to express numbers verbally!

However, in a scholarly context, scientific notation is often used in papers and studies to express a variety of numbers, especially when there are multiple numbers of a similar type being compared. The following section shows how scientific notation can be used to compare numbers.

Comparing numbers using scientific notation

One of the benefits of scientific notation is that it allows us to compare the relative size of quantities that are very large or very small. This is especially clear when the quantities are different orders of magnitude; that is, when they have different exponents in scientific notation. Using scientific notation can help make this more obvious than ordinary notation.

Example \(\PageIndex{4}\)

Which is larger? \(89000000\) or \(9800000\)?

Solution

The most likely way to approach a problem written in this way is to squint at the numbers really hard, and count how many zeroes there are either with your eyes or a tool like a pencil. If you do that, you'll see that there are \(6\) zeroes in the first number and only \(5\) in the second number, so the first number must be larger. However, if we had been given these numbers in scientific notation: \[8.9 \times 10^{7} \text{ or } 9.8 \times 10^{6}\] then it is very clear that the first number is larger, since it is multiplied by a larger power of \(10\). As a side note, these would be considered quantities with different orders of magnitude; \(89000000\) is one order of magnitude larger than \(9800000\).

How might this be useful in context? Consider the following example.

Example \(\PageIndex{5}\)

 

Many chemicals are harmful if ingested, inhaled, or otherwise consumed by humans. Often, we use blood concentrations of chemicals to define the quantity of a substance that is considered harmful. Mercury and lead are two chemicals known to be harmful at very low levels. Lead can be shown to cause harm at concentrations greater than \(3.5 \times 10^{-6}\) grams per deciliter of blood, whereas mercury can be shown to cause harm at concentrations greater than \(1 \times 10^{-7}\) grams per deciliter of blood.

Use the information above to compare the effects of equal amounts of lead and mercury exposure.

 

Solution

There are many ways to answer this question, of course. The point of this example, and of similar exercises you will find throughout the text, is to practice reading, analyzing, and describing quantitative information in context.

What we see in the paragraph above is that while lead and mercury are both dangerous to humans, even at small quantities, a small amount of mercury is more harmful than the same amount of lead, since the concentration required for harm is lower than that of lead.

For example, someone with \(1.5 \times 10^{-7}\) grams of mercury per deciliter of blood would be above the level of toxicity for mercury, but someone with the same amount of lead — \(1.5 \times 10^{-7}\) grams of lead per deciliter of blood — would be below the level of toxicity for lead, since \(1.5 \times 10^{-7}\) is less than \(3.5 \times 10^{-6}\).

The use of scientific notation allows us to compare these two numbers more clearly than if they were written in standard notation.

And of course, please note: both mercury and lead, at any level, can be very dangerous, and represent serious environmental hazards for many. Please avoid these chemicals as much as possible.

A note about notation

The expression of scientific notation above is common in published textbooks and journals in a variety of fields. However, technology such as spreadsheets, calculators, and computation programs use various shorthand for scientific notation. There are many variations on this as various manufacturers have adopted different conventions, but a fairly common notation is the use of "E" to indicate the power of \(10\) in an expression in scientific notation. For example, your computer might display the number \(587,000,000\) as "5.87E8," which is shorthand for \(5.87 \times 10^8\). Be aware of this when you are using technology to handle very large or very small numbers!

Exercises

1. Convert \(792,000,000,000\)  to scientific notation.
2. Convert \(.0000508\)  to scientific notation.
3. Convert \(8.6 \times 10^7\) to ordinary notation.
4. Convert \(7.2 \times 10^{-9}\) to ordinary notation.
5. Which is larger: \(3.02 \times 10^5\) or \(4.02 \times 10^4\)? Explain how you know.
6. Which is larger: \(6.11 \times 10^{-15}\) or \(1.001 \times 10^{-13}\)? Explain how you know.
7. Find a recent news article (from the last 6 months) that contains a very large or very small number. (For the purposes of this exercise, "very large" means greater than \(1\) million, and "very small" means less than \(1\) one-millionth.) For that number: 
   1. Write the number in ordinary notation. 
   2. Write the number in scientific notation. 
   3. Of the types of notation — including how the number was originally presented in the article — write a 1-2 sentence reflection on which type of notation you believe most clearly expresses the size of the number and why. 
   4. Include a link to your news article.