What is Scientific Notation?

Scientists (imagine that) and others often need to perform calculations requiring them to use extremely large and extremely small numbers. These extreme numbers have many decimal places, tons of zeros, and take a long time to write accurately. Someone noticed that these extreme numbers are still decimal (base ten) numbers. We know that decimal number place values are powers of ten, so why not write merely the useful part of the number times 10 to a power that replaces all those zeros? Even better, write exactly one non-zero digit (you know: 1, 2, 3, 4, 5, 6, 7, 8, or 9) to the left of the decimal point and all the other useful digits to the right, then a multiplication sign, then 10 with a positive or negative exponent.

Example 1

Write 357,000,000,000 in Scientific Notation Form:

What to do: find the first digit that it not a zero. What is it? Yep, a 3.
Write the 3, then a decimal point, and the other digits until you get to the zeros. What do we have now? 3.57

Next, write X or • and 10: 3.57 X 10

Now, we count the number of decimal places we need to move the decimal from behind the 3 to its original place. We need to move the Scientific Notation Form decimal 11 places to the right to get it back to its original place (9 for the zeros and one each for the 5 and the 7).

REMEMBER: we are not moving the original decimal point. We are moving the Scientific Notation Form decimal point!!!!

What does the number look like now: 3.57 • 10¹¹

Example 2

Write 0.00000000625 in Scientific Notation form:
Write the first non-zero digit, then a decimal point, then the rest of the useful digits: 6.25
Now, write 6.25 times 10 to a power: 6.25 X 10^?

But what power should we use? We first need to know which way we are going to need to move the decimal point to get it back to its original position, right or left? In this number, we need to move the decimal back to the left. Left is a negative direction, so our exponent is negative. We also need to go 9 places to the left, so our number is: 6.25 X 10^-9.

Okay, all well and good. But what can we do with these Scientific Notation numbers?

We can multiply and divide with them using the Rules for Exponents, especially the Product and Quotient Rules.

Example 3

Multiply 3.57 • 10¹¹ times 6.25 X 10^-9 and write in Scientific Notation form.

Use your calculator to multiply 3.57 times 6.25: 22.3125

Use the power rule to multiply 10¹¹ times 10^-9: 11 + -9 = 2.

Is 22.3125 X 10² in Scientific Notation form? No! What do we need to do?

We need exactly one non-zero digit left of the decimal point, the other useful digit to the right, then X 10^power. So, we write 2.23 X 10³, because we need to move the decimal point an additional one place to the right to get it back to its original position ( the exponent of 2 that we have plus the one more we need to move the decimal place). (Notice: we also rounded the number to 2 decimal places. Check your directions for the number of places you are required to have in your result.)

Can we multiply Scientific Notation numbers on our calculators?

Certainly. Set the MODE to Sci: [MODE], [ > ], [Enter]
Sci will be highlighted, Normal and Eng will be clear.

In the calculation window, type [ 3 ] [ . ] [ 5 ] [ 7 ] [ 2nd] [ , ] [ 1 ] [ 1 ]
You should see 3.57E11 The [2nd] [ , ] gives you the E {look for EE above the comma key}.

NOW type [ X ], because we are multiplying, then [ 6 ] [ . ] [ 2 ] [ 5 ] [2nd] [ , ] [ (–) ] [ 9 ] [ENTER]. The calulator displays 2.23125E3.
How do we divide using Scientific Notation?

Example 4

Divide 3.57 X 10¹¹ by 6.25 X 10^–9 and write your quotient in Scientific Notation form.

3.57 ÷ 6.25 = .5712. To find the exponent, we use the Quotient Rule. 11 – -9 = 20.

Wait! .5712 X 10²⁰ is NOT in Scientific Notation!

In Scientific Notation, we need only the 5 to the left of the decimal point: 5.712
This means we need to move our Scientific Notation decimal one place to the left to return it ot its original position. Since left means negative, we subtract one from the exponent of 20. Our quotient in Scientific Notation form is 5.725 X 10¹⁹.

Let’s use our calulators to check:
[CLEAR]
[ 3 ] [ . ] [ 5 ] [ 7 ] [2nd] [ , ] [ 1 ] [ 1 ] [ ÷ ] [ 6 ] [ . ] [ 2 ] [ 5 ] [2nd] [ , ] [ (-) ] [ 9 ]

Your screen should read: 3.57E11/6.25E-9
[ENTER]

Now, your screen should read:

| 3.57E11/6.25E-9 |
| 5.712E19 |