Scientific Notation

Very often, scientists are working with numbers that are large (or small) beyond comprehension. For example, here's the mass of the sun.

1988500000000000000000000000000 kg1988500 000000 000000 000000 000000 \text{ kg}

Now, don't you think it would be too cumbersome a number to both read and write as such? Just imagine counting all those zeros. Then, the space it would need.

And remember, it's not even close to the largest numbers we have to deal with.

So, there was a need for system that would make expressing such numbers easier for us — without losing any information in the process. And that's where the scientific notation comes in.

What Is the Scientific Notation?

Scientific notation refers to a system of writing numbers based on the following rules.

  • Each number is written as a product of two parts.
  • The first part is a decimal number with an absolute value between 1\hspace{0.2em} 1 \hspace{0.2em} and 10\hspace{0.2em} 10 \hspace{0.2em} (can be 1\hspace{0.2em} 1 \hspace{0.2em} but must be less than 10\hspace{0.2em} 10 \hspace{0.2em}).
  • The second part is an exponent of 10\hspace{0.2em} 10 \hspace{0.2em} written as 10n\hspace{0.2em} 10^n \hspace{0.2em}, where n\hspace{0.2em} n \hspace{0.2em} is some integer.

For example, the mass of the sun in the scientific notation would be 1.9885×1030 kg\hspace{0.2em} 1.9885 \times 10^{30} \text{ kg} \hspace{0.2em}.

So, the first part, 1.9885\hspace{0.2em} 1.9885 , is a decimal number between 1\hspace{0.2em} 1 \hspace{0.2em} and 10\hspace{0.2em} 10. The second part is 1030\hspace{0.2em} 10^{30}.

Note — In the UK, scientifc notation is often referred to as "scientific form" or "standard form".

How to Convert to Scientific Notation

The steps we take to convert a number to the scientific notation vary depending on whether or not the number is less than 1\hspace{0.2em} 1 \hspace{0.2em} (in absolute value). So let's look at the different cases one by one.

Less Than 1

Example
Express 0.03705\hspace{0.2em} 0.03705 \hspace{0.2em} in scientific notation.

Solution

Step 1.  Count the number of zeros between the decimal point and the first non-zero digit. Add 1\hspace{0.2em} 1 \hspace{0.2em} to it. We'll call it n\hspace{0.2em} {\color{Red} n} .

Here, we have 1\hspace{0.2em} {\color{Teal} 1} \hspace{0.2em} zero between the decimal point and the first non-zero digit (3)\hspace{0.2em} (3) \hspace{0.2em}. So,

n=1+1=2\begin{align*} {\color{Red} n} \hspace{0.25em} &= \hspace{0.25em} {\color{Teal} 1} + 1 \\[1em] &= \hspace{0.25em} {\color{Red} 2} \end{align*}

Step 2.  Rewrite the decimal number moving the decimal point to the right of the first non-zero digit. Also, get rid of any leading zeros before the decimal point.

0.03705=3.705×0.03705 \hspace{0.25em} = \hspace{0.25em} 3.705 \times \rule{1.5em}{0.05em}

Step 3.  For the second part, use 10−n\hspace{0.2em} 10^{- {\color{Red} n} } \hspace{0.2em}.

In step 1, we got n=2\hspace{0.2em} {\color{Red} n} = {\color{Red} 2} \hspace{0.2em}. So,

0.03705=3.705×10−20.03705 \hspace{0.25em} = \hspace{0.25em} 3.705 \times 10^{- {\color{Red} 2} }

That's it.

Greater Than Or Equal to 1

Example
Express 56412.8\hspace{0.2em} 56412.8 \hspace{0.2em} in scientific notation.

Solution

Step 1.  Count the number of digits before the decimal point. If the number doesn't have a decimal point, take the total number of digits in it.

Subtract 1\hspace{0.2em} 1 \hspace{0.2em} from it. We'll call it n\hspace{0.2em} {\color{Red} n} .

Here, we have 5\hspace{0.2em} {\color{Teal} 5} \hspace{0.2em} digits before the decimal point. So,

n=5−1=4\begin{align*} {\color{Red} n} \hspace{0.25em} &= \hspace{0.25em} {\color{Teal} 5} - 1 \\[1em] &= \hspace{0.25em} {\color{Red} 4} \end{align*}

Step 2.  Rewrite the given number moving the decimal point to the right of the first digit.

56412.8=5.64128×56412.8 \hspace{0.25em} = \hspace{0.25em} 5.64128 \times \rule{1.5em}{0.05em}

Step 3.  For the second part, use 10n\hspace{0.2em} 10^ {\color{Red} n} \hspace{0.2em}.

In step 1, we got n=4\hspace{0.2em} {\color{Red} n} = {\color{Red} 4} \hspace{0.2em}. So,

56412.8=5.64128×10456412.8 \hspace{0.25em} = \hspace{0.25em} 5.64128 \times 10^ {\color{Red} 4}
Example
Express 2\hspace{0.2em} 2 \hspace{0.2em} in scientific notation.

Solution

Step 1.  There's no decimal point in the number, so we take the total number of digits, 1\hspace{0.2em} 1 \hspace{0.2em}.

Subtracting 1\hspace{0.2em} 1 \hspace{0.2em} from it, we get —

n=1−1=0\begin{align*} {\color{Red} n} \hspace{0.25em} &= \hspace{0.25em} {\color{Teal} 1} - 1 \\[1em] &= \hspace{0.25em} {\color{Red} 0} \end{align*}

Step 2.  Generally, we would rewrite the given number moving the decimal point to the right of the first digit (so 2.\hspace{0.2em} 2. \hspace{0.2em}). But don't want a decimal point at the end in scientific notation, so we can leave it as 2\hspace{0.2em} 2 \hspace{0.2em} itself.

2=2×2 \hspace{0.25em} = \hspace{0.25em} 2 \times \rule{1.5em}{0.05em}

Step 3.  For the second part, we use 10n\hspace{0.2em} 10^ {\color{Red} n} \hspace{0.2em}. So 100\hspace{0.2em} 10^ {\color{Red} 0} \hspace{0.2em}.

2=2×1002 \hspace{0.25em} = \hspace{0.25em} 2 \times 10^ {\color{Red} 0}
Example
Express −2569\hspace{0.2em} -2569 \hspace{0.2em} in scientific notation.

Solution

Step 1.  The number has 4\hspace{0.2em} 4 \hspace{0.2em} digits. Subtracting 1\hspace{0.2em} 1 \hspace{0.2em}, we get —

n=4−1=3\begin{align*} {\color{Red} n} \hspace{0.25em} &= \hspace{0.25em} {\color{Teal} 4} - 1 \\[1em] &= \hspace{0.25em} {\color{Red} 3} \end{align*}

Step 2.  Lets rewrite the given number moving the decimal point to the right of the first digit.

−2569=−2.569×-2569 \hspace{0.25em} = \hspace{0.25em} -2.569 \times \rule{1.5em}{0.05em}

Step 3.  For the second part, use 10n\hspace{0.2em} 10^ {\color{Red} n} \hspace{0.2em}.

In step 1, we got n=3\hspace{0.2em} {\color{Red} n} = {\color{Red} 3} \hspace{0.2em}. So,

−2569=−2.569×103-2569 \hspace{0.25em} = \hspace{0.25em} -2.569 \times 10^ {\color{Red} 3}

Scientific Notation to the Usual Form

To convert a number in scientific notation to decimal notation —

Take the exponent of 10\hspace{0.2em} 10 \hspace{0.2em} and move the decimal point right if the exponent is positive (left, if it is negative) by as many places.

Append 0\hspace{0.2em} 0s to the appropriate end of the number if you run out of digits (one 0\hspace{0.2em} 0 \hspace{0.2em} for each missing digit).

Let me explain using examples.

Example

Express 1.83×106\hspace{0.2em} 1.83 \times 10^6 \hspace{0.2em} in decimal notation.

Solution

Here, the exponent is 6\hspace{0.2em} 6 \hspace{0.2em} (positive) and so we need to move the decimal point 6\hspace{0.2em} 6 \hspace{0.2em} places to the right.

However, we have only 2\hspace{0.2em} 2 \hspace{0.2em} digits to the right of the decimal point. We would need four additional digits. So we append four 0\hspace{0.2em} {\color{Red} 0} s.

1.83×106=18300001.83 \times 10^6 = 183 {\color{Red} 0000}

Done.

Example

Express 2.569×10−2\hspace{0.2em} 2.569 \times 10^-2 \hspace{0.2em} in decimal notation.

Solution

This time, the exponent is −2\hspace{0.2em} -2 \hspace{0.2em} and so we need to move the decimal point 2\hspace{0.2em} 2 \hspace{0.2em} places to the left (because the exponent is negative).

However, we have only 1\hspace{0.2em} 1 \hspace{0.2em} digits to the left of the decimal point. We would need one additional digits. So we append a 0\hspace{0.2em} {\color{Red} 0} \hspace{0.2em} there.

2.569×106=.02569=0.02569\begin{align*} 2.569 \times 10^6 \hspace{0.25em} &= \hspace{0.25em} . {\color{Red} 0} 2569 \\[1em] &= \hspace{0.25em} 0. {\color{Red} 0} 2569 \end{align*}

That's our answer.

Working With Numbers In the Scientific Notation

Alright, let's look at how we can do the four basic operations on numbers in the scientific notation.

When it comes to doing operations, there's nothing special about numbers in the scientific notation. We work with them just like we would with any instance of a decimal number and an exponent multiplied together.

Multipying/Dividing

To multiply/divide numbers in the scientific notation,

  • Multiply/divide the first parts of the numbers (parts with the decimal numbers).

  • Take the sum of the exponents when multiplying (and difference when dividing).

  • If necessary, make sure the result is in scientific notation.

Again, let me explain using the following examples.

Example
Find the product of 2×102\hspace{0.2em} 2 \times 10^2 \hspace{0.2em} and 3.15×104\hspace{0.2em} 3.15 \times 10^4 \hspace{0.2em}.

Solution

Let's start by rewriting the problem in a more mathematical way.

2×102×3.15×1042 \times 10^2 \hspace{0.25em} \times \hspace{0.25em} 3.15 \times 10^4

And now let's rearrange things a bit so it reflects what we want to do. I'll also color similar parts that we'll process together.

2×3.15×102×104 {\color{Red} 2} \times {\color{Red} 3.15} \times {\color{Teal} 10^2} \times {\color{Teal} 10^4}

Great! Next, as mentioned above, we multiply the decimal numbers together and take the sum of the exponents. So,

2×3.15×102×104=6.30×102+4=6.3×106\begin{align*} & {\color{Red} 2} \times {\color{Red} 3.15} \times {\color{Teal} 10^2} \times {\color{Teal} 10^4} \\[1em] = \hspace{0.5em} & {\color{Red} 6.30} \times {\color{Teal} 10^{2 + 4}} \\[1em] = \hspace{0.5em} & {\color{Red} 6.3} \times {\color{Teal} 10^6} \end{align*}

And there's our answer.

Example
Simplify : 6.1×102×3.4\hspace{0.2em} 6.1 \times 10^2 \times 3.4 \hspace{0.2em}

Solution

Compared to the last example, here we have a couple of differences.

To begin with, the second number is not in the scientific notation. But as you'll see, it doesn't make any real difference to how we proceed here. Just that we won't need to add the exponents.

So we'll follow the same steps as in the previous one and see how far it takes us.

6.1×102×3.4=6.1×3.4×102=20.74×102\begin{align*} 6.1 \times 10^2 \times 3.4 \hspace{0.5em} = \hspace{0.5em} & {\color{Red} 6.1} \times {\color{Red} 3.4} \times {\color{Teal} 10^2} \\[1em] = \hspace{0.5em} & {\color{Red} 20.74} \times {\color{Teal} 10^2} \end{align*}

Here comes the other difference. Our product isn't in scientific notation. But we already know how to convert it into scientific notation.

20.74×102=2.074×10320.74 \times 10^2 \hspace{0.5em} = \hspace{0.5em} 2.074 \times 10^3
Example
Simplify :
4.2×1078.4×104\hspace{1em}\frac{4.2 \times 10^7}{8.4 \times 10^4}

Solution

This time, we have a division problem. No worries. Everything remains the same, except that we take the difference of the exponents. So,

4.2×1078.4×104=4.28.4×107104=0.5×107−4=0.5×103\begin{align*} \frac{4.2 \times 10^7}{8.4 \times 10^4} \hspace{0.5em} = \hspace{0.5em} &\frac{ {\color{Red} 4.2} }{ {\color{Red} 8.4} } \times \frac{ {\color{Teal} 10^7} }{ {\color{Teal} 10^4} } \\[1.5em] = \hspace{0.5em} & {\color{Red} 0.5} \times {\color{Teal} 10^{7-4}} \\[1.5em] = \hspace{0.5em} & {\color{Red} 0.5} \times {\color{Teal} 10^3} \end{align*}

Here again, the product in not in scientific notation. So, let's do that.

0.5×103=5×1020.5 \times 10^3 \hspace{0.5em} = \hspace{0.5em} 5 \times 10^2

Adding/Subtracting

How we add (or subtraction) numbers in scientific notation depends on whether their exponent parts are the same or not.

It will be easier to explain using examples. So let's dive in.

When Exponents Are the Same
Example
Simplify : 4.5×102+5.7×102\hspace{0.2em} 4.5 \times 10^2 + 5.7 \times 10^2 \hspace{0.2em}

Solution

In this example, the exponent parts are the same 102\hspace{0.2em} 10^2 \hspace{0.2em}.

And in such cases, we —

  1. add together the decimal number parts, and

  2. copy over the exponent part.

So let's apply these steps to the problem on hand.

4.5×102+5.7×102=(4.5+5.7)×102=10.2×102\begin{align*} &4.5 \times {\color{Red} 10^2} + 5.7 \times {\color{Red} 10^2} \\[1em] = \hspace{0.3em} &(4.5 + 5.7) \times {\color{Red} 10^2} \\[1em] = \hspace{0.3em} &10.2 \times {\color{Red} 10^2} \end{align*}

And we've got the sum. Now, it's not in scientific notatio. So, let's do that.

10.2×102=1.02×10310.2 \times 10^2 \hspace{0.3em} = \hspace{0.3em} 1.02 \times 10^3
When Exponents Are Not the Same
Example
Simplify : 3.715×104−1.9×102\hspace{0.2em} 3.715 \times 10^4 - 1.9 \times 10^2 \hspace{0.2em}

Solution

When the exponents are not the same, we first need to make them the same. We can convert all the numbers to the usual form. Or pick one of the exponents and make them all equal to that exponent.

Here's what I mean.

3.715×104−1.9×102=37150−190=36960\begin{align*} &3.715 \times 10^4 - 1.9 \times 10^2 \\[1em] = \hspace{0.5em} &37150 - 190 \\[1em] = \hspace{0.5em} &36960 \end{align*}

Once we have the difference, we can go back to the scientific form.

36960=3.696×10436960 \hspace{0.3em} = \hspace{0.3em} 3.696 \times 10^4

Alternate approach — Instead of converting the numbers to the usual form, we could have made the exponent for each number the same. Say 2\hspace{0.2em} 2 \hspace{0.2em}.

3.715×104−1.9×102=371.5×102−1.9×102=(371.5−1.9)×102=369.6×102\begin{align*} &3.715 \times 10^4 - 1.9 \times 10^2 \\[1em] = \hspace{0.5em} &371.5 \times {\color{Red} 10^2} - 1.9 \times {\color{Red} 10^2} \\[1em] = \hspace{0.5em} &(371.5 - 1.9) \times {\color{Red} 10^2} \\[1em] = \hspace{0.5em} &369.6 \times {\color{Red} 10^2} \end{align*}

Again, returning to scientific notation, we have,

369.6×102=3.696×104369.6 \times 10^2 \hspace{0.3em} = \hspace{0.3em} 3.696 \times 10^4

And with that, we come to the end of this elementary tutorial on absolute value. Until next time.