Scientific notation

Scientific notation is an essential tool in both mathematics and science. You may already know the rules and uses, but we’ll start with the basics so you can use it confidently.

Scientific notation has three parts:

1. Your simplified number

2. $\times$ (multiplied by)

3. $10$ to some power

Scientific notation is practical because it makes very large and very small numbers easier to read and write. It does not change the value of a number - it only changes how the number is written.

Consider the number $15$. This isn’t a number you need scientific notation for, but it’s a simple way to see the idea.

$15$ written in scientific notation is $1.5\times 10^1$. Here’s why: $10^1=10$, and $1.5\times 10=15$.

In this chapter, you’ll focus on converting to and from scientific notation. The process stays the same no matter what number you start with.

Converting from standard to scientific notation

The goal is to rewrite your number so there is exactly one digit to the left of the decimal point. There can be several digits to the right of the decimal, but in these examples we’ll round so there is one digit to the right.

To convert, you’ll move the decimal point and count how many places it moves.

Take this example: $12,345,678$ is a large number. In scientific notation, it should look like something close to $1.2$ multiplied by a power of $10$.

1. View the whole number with a decimal point: $12,345,678.0$

2. Move the decimal one place to the left (this divides the number by $10$): $1,234,567.80$

3. Keep moving the decimal left until there is only one digit to the left of the decimal: $1.23456780$

4. Count how many places you moved the decimal (how many groups of ten were removed): you moved it $7$ places.

5. Round the number so there is one digit on each side of the decimal: $1.2$

6. Since you removed $7$ groups of $10$, attach “$\times 10^7$” to your simplified number: $1.2\times 10^7$

7. Nice work! You just converted to scientific notation!

Very small numbers can also be written in scientific notation. The process is similar, but there’s one key rule: the digit to the left of the decimal must be non-zero.

Use the number $0.000437$.

1. We already see the decimal, so we can skip the usual first step.

2. Move the decimal one place to the right (this multiplies the number by $10$): $0.00437$

3. Keep moving the decimal right until there is one non-zero digit to the left of the decimal point: $4.37$

4. Count how many places you moved the decimal (how many groups of ten were added): you moved it $4$ places.

5. Round so there is one digit on each side of the decimal: $4.4$.

6. Since you added $4$ groups of $10$ to get $4.4$, you must multiply by a negative power of $10$ to keep the value the same: $4.4\times 10^{-4}$

7. The negative exponent is important: it ensures the decimal would move back to the left if you converted back to standard form.

8. If you’re unsure about the sign of the exponent, multiply it out to check. You didn’t change the number, so it should match the original. For example, $4.4\times 10^{-4}$ gives $0.000437$.

Count your decimal places:

• For large numbers, moving the decimal left gives a positive exponent.
• For small numbers, moving the decimal right gives a negative exponent.

Converting from scientific to standard notation

Converting from scientific notation back to standard notation is the reverse process. You carry out the multiplication shown by the expression.

For example, consider $2.6\times 10^3$.

1. Multiply $2.6$ by $10^3$.

2. That’s it! With a calculator, you can enter the expression directly.

3. Without a calculator, you can use the exponent to decide how to move the decimal point.

4. On a calculator, $2.6\times 10^3=2600$.

5. Without a calculator: $10^3=1000$, so $2.6\times 1000=2600$.

6. You can also think of this as moving the decimal point:

   • If the exponent is positive, move the decimal that many places to the right.
   • If the exponent is negative, move the decimal that many places to the left.
   • Fill any gaps with $0$s.

Addition and subtraction in scientific notation

The easiest way to do arithmetic with scientific notation is to use a calculator. Still, it helps to understand what the calculator is doing.

Addition and subtraction can be tedious in scientific notation. A reliable method is:

• Convert each number to standard form.
• Add or subtract.
• Convert the result back to scientific notation.

Make sure you’re comfortable with the conversions above so this method feels straightforward.

Try this example on your own:

What is the sum of the following expression?

$$(5.1\ast 10^8)+(6.5\ast 10^8)$$

You’ll notice that adding in standard form increased the number of digits by one, so the power of ten increased from $10^8$ to $10^9$. Then the simplified number was rounded to one decimal place.

Multiplication and division in scientific notation

For multiplication, you can convert everything to standard form (as you did for addition), or you can multiply directly in scientific notation.

To multiply directly, treat scientific notation as two parts:

• the simplified number
• the power of ten

Then multiply those parts separately.

What is the product of the following expression?

$$(2.4\ast 10^8)\ast (8.9\ast 10^3)$$

First multiply the simplified numbers: $2.4\ast 8.9$.

This product is $21.4$ (rounded). Notice that $21.4$ has two digits before the decimal, so it is not in proper scientific notation. To fix that, move the decimal one place to the left to get $2.14$, which increases the exponent by $1$.

Now handle the powers of ten:

$$10^8\ast 10^3$$

Since the bases are the same ($10$), add the exponents: $10^{8+3}=10^{11}$. This will be discussed more in the section on [Elementary Algebra].

Putting it together:

• simplified number: $2.1$ (rounded to one decimal place after adjusting)
• powers of ten: $10^{11}$
• adjustment from moving the decimal: increase the exponent by $1$

So the final answer is $2.1\ast 10^{12}$.

Division works the same way, except you divide the simplified numbers and subtract exponents. One common difference is that you may need to move the decimal to the right to get back to proper scientific notation, which subtracts $1$ from the exponent (you can think of this as “carrying a negative one”).

If you were to perform the above example using division instead of multiplication the final answer would be $2.7\ast 10^4$. Can you get to that answer?

One last note!

You can also enter scientific notation directly on many calculators. Look for an $EE$ button. On the TI-83 and TI-84, press “2nd” and then the comma $,$. The $EE$ stands for “$\times 10^x$.”

For example, typing $1.25\times 10^7$ would look like $1.25EE7$ (many screens show a single capital E instead of both). If you have a different calculator, look up how it enters scientific notation and practice with a few examples.

Key points

Format. Scientific notation is always comprised of two parts: the simple number and the exponent ($10$ to some power)

Converting standard to scientific. Count how many places you must move the decimal point, and that number becomes your exponent of $10$ in scientific notation. Moving the decimal left makes the exponent positive, moving it right makes it negative

Converting scientific to standard. The exponent above $10$ shows you how many decimal places to move your decimal point. Positive means move it back to the right that many spaces (since it originally came from the right), negative means move it back to the left that many spaces (since it originally came from the left).

Addition/subtraction. The most efficient way is always with your calculator! Otherwise, convert scientific numbers to standard, add or subtract them, then convert them back.

Multiplication/division. The most efficient way is always with your calculator! Otherwise, separate the two parts of the scientific numbers into the simple number and the $10$ exponent, multiply or divide them separately, then combine them at the end (remember to carry any remainders!)