# Scientific Notation

Scientific notation is a way to concisely write numbers which are very large (which often happens in astronomy) or very small (which often happens in cell biology). In this lesson, we will learn how to use scientific notation, including how to do arithmetic with numbers in that form.

## Scientific notation

Often scientists find themselves needing to deal with very large numbers. For example, the mass of the sun is roughly:

$$1,989,000,000,000,000,000,000,000,000,000\;\text"kilograms"$$

We would like to find a simpler way of writing very large numbers like this.

Compute each of the numbers in the table below.

If $n$ is a positive integer, $10^n$ can be written as 1 followed by $n$ zeroes. So multiplying a number by $10^n$ just shifts the decimal point in that number $n$ places to the right. If the result is large, this can make it look a lot simpler. For example, we could say that the mass of the sun is roughly:

$$1.989 ⋅ 10^30\; \text"kilograms"$$

which is simpler than the first way we wrote it.

 What is $35 ⋅ 10^2$?
 What is $3.5 ⋅ 10^3$?

Notice that there can be different ways of writing the same number as a multiple of a power of 10. We’ll pick out one specific way:

If a number is written as $a ⋅ 10^n$ for some integer $n$, and $1 ≤ {|a|} < 10$ (that is, $a$ has exactly 1 digit before the decimal point, and that digit is nonzero), then the number $a ⋅ 10^n$ is written in scientific notation.

For example, 2600 can be written in scientific notation as $2.6 ⋅ 10^3$.

 Write $18000$ in scientific notation.
 Write $490$ in scientific notation.

Write $58 ⋅ 10^3$ in scientific notation.

 Write $35 ⋅ 10^2$ in scientific notation.

We can write numbers which are between 0 and 1 in scientific notation also, by raising 10 to a negative power. Note that $10^{-2}=1/10^2=1/100=0.01$.

 What is $3.8 ⋅ 0.01$?
 What is $10^{-3}$?
 What is $5.2 ⋅ 10^{-3}$?
 Write 0.0025 in scientific notation.
 Write $0.43 ⋅ 10^{-4}$ in scientific notation.

We usually write numbers which are very large or very small in scientific notation. With other numbers, for instance between 0.01 and 100, we usually do not, even in a scientific context.

## Mass of a planetary system

The mass of the dwarf planet Pluto is $1.305 ⋅ 10^22$ kg. (The abbreviation “kg” stands for “kilograms.” On Earth, one kilogram weighs about 2.2 pounds.) The mass of its moon Charon is $1.59 ⋅ 10^21$ kg. We would like to add these two numbers together to find the total mass of the Pluto-Charon system. (Pluto has other moons, but they are much smaller, so we will ignore them.)

To start with, we need to write both of these masses using the same exponent. What is the mass of Pluto, in units of $10^21$ kg?

On the grid to the left, each small green square represents a mass of $10^21$ kg. The left-hand group of small green squares shows the mass of Pluto, while the right-hand group shows the mass of Charon.

What is the total mass of Pluto and Charon, in units of $10^21$ kg?

 What is the total mass of Pluto and Charon, in scientific notation?

The total mass of the Pluto-Charon system is $1.464 ⋅ 10^22$ kg. Since the mass of Charon is $1.59 ⋅ 10^21$ kg, Charon makes up

$$\table {1.59 ⋅ 10^21\,\text"kg"}/{1.464 ⋅ 10^22\,\text"kg"},=,1.59 / 1.464 ⋅ 10^21 ⋅ 10^{-22}; ,≈, 1.09 ⋅ 10^{-1}; ,=, 0.109$$

of the mass of the entire system.

What fraction of the mass of the entire system does Pluto make up? Remember that the mass of Pluto is $1.305 ⋅ 10^22$ kg. Round your intermediate answer to three decimal places.

The total mass of the Earth and the Moon is $6.045 ⋅ 10^24$ kg. The mass of the Moon by itself is $7.348 ⋅ 10^22$ kg. What fraction of the mass of the entire Earth-Moon system does the Moon make up? Round your intermediate answer to three decimal places.

Click to see the masses of the Earth and the Moon pictured to the left.

The total mass of the Jupiter system (Jupiter and all its moons) is $1.898 ⋅ 10^27$ kg. The mass of Jupiter’s largest moon Ganymede is $1.482 ⋅ 10^23$ kg. What fraction of the mass of the entire Jupiter system does Ganymede make up? Round your intermediate answer to three decimal places, and write your final answer in scientific notation.

## Number of cells in the human body

A picture of some animal skin cells, as seen under a microscope, is shown to the left. You could take a sample of your own skin cells and look at it under a microscope, and then measure the size of one of those cells. We would like to try to come up with a rough estimate for the total number of cells in the human body, by assuming that your skin cells are typically sized for human cells.

You measure each human skin cell to be about $3 ⋅ 10^{-5}$ m (meters) in length. They look like they are roughly cubical. If we assume that they are actually cubes, what is the volume of each cell? Write your final answer in scientific notation.

We would like to find the mass of a human skin cell, which can be done by multiplying its volume by its density. Since human bodies are mostly water, we will assume that the cell has the same density as water: 997 kilograms per cubic meter (997 kg$∕$m$^3$). Using this information, estimate the mass of a skin cell. Write your final answer in scientific notation, rounded to one decimal place.

The mass of a typical adult human body is roughly 65 kg. If we divide this total mass by the mass of an average single cell, we’ll get the number of cells in an entire human body. Assuming that skin cells have average mass, how many cells are in a typical adult human’s body? Round your intermediate answer to one decimal place, and write your final answer in scientific notation.

Biologists have made a more careful count of the number of cells in the human body, and determined that it is about $3.72 ⋅ 10^13$. What is the ratio of this correct count to the rough estimate we made? Round your intermediate answer to two decimal places.

 Are skin cells larger or smaller than average human cells?