SCIENTIFIC NOTATION

Introduction Scientists like us often use numbers that are really, really big. For instance, the mass of the Earth is about 5970000000000000000000000 kg. We also use really small numbers like the mass of a single electron, which is 0.000000000000000000000000000000911 kg. There are problems with writing numbers like that. Foremost, it takes a lot of time to write down all those zeros, and make sure you haven't forgotten any. Lose one zero and you've messed up big time! To avoid these difficulties, we use scientific notation.

Let's look at an example. In scientific notation, Earth's mass is 5.97×10²⁴ kg. The 5.97 part is called the coefficient; the 24 is called the exponent. Remember, an exponent simply describes how many times you are multiplying a number by itself. 2³ is 2×2×2 which is 8 and 2⁴ is 2×2×2×2 or 16. It's especially easy when multipling 10's:

10³ is 10 × 10 × 10 = 1000, or a one with three zeros (a thousand).
10⁶ is 10 × 10 × 10 × 10 × 10 × 10 = 1000000, a one followed by six zeros (a million).

So 10²⁴ is a 1 followed by twenty-four zeros, 1000000000000000000000000. The mass of the Earth (in kilograms) is 5.97 times bigger than that, so that's how we write it: 5.97×10²⁴ kg. If you wanted to write that out not-in-scientific-notation, simply write 5.97, and then move the decimal point 24 places to the right, filling in zeros to hold the places:

Small numbers are trickier; we have to use negative exponents. Recall that a negative exponent means the inverse ("1 divided by...") of the number raised to the exponent. For instance:

2^-3 = 1/2³ = 1/8
10^-3 = 1/10³ = 1/1000 = 0.001 (one thousandth)
10^-6 = 1/10⁶ = 1/1000000 = 0.000001 (one millionth)

The mass of a proton, in scientific notation, is 1.67×10^-27kg. To write this out not-in-scientific-notation, write 1.67 and then move the decimal point to the left 27 places, filling in zeros as needed.

The coefficient and exponent are always written as decimal numbers, not as fractions. (For example, 1.4×10², rather than 7/5×10².)

In general, we prefer to keep the coefficient between 1 and 10 (either positive or negative) by adjusting the exponent. For example, we could write the mass of a proton as 16.7×10^-28 kg, or 167×10^-29 kg – mathematically they are all the same! But we prefer to write it as 1.67×10^-27 kg. You don’t need to do this for intermediate results when you’re doing a long calculation, just for ‘final answers’.

Activities & Practice
to do as you read

1. The mass of an electron is 9.11x10^-31kg.
What is the coefficient? ______
What is the exponent? ______

Fill in each blank with the word POSITIVE or NEGATIVE.
2. Large numbers have ______ exponents.
3. Small numbers have ______ exponents.

4. Write 9.11x10^-31kg as a regular decimal number.

5. How many seconds are there in an hour? How many in a day? Write these in scientific notation.

6. What is your age in years, in scientific notation?

7. A light-year is a distance equal to 9.461x10¹⁵meters. Write this as a regular number (i.e. not in scientific notation).

8. In scientific notation, what is (a) one thousand, (b) one million, (c) one billion, (d) one trillion, (e) one thousandth, (f) one millionth, (g) one billionth, (h) one trillionth

9. What are these numbers, not-in-scientific-notation:
   (a) 3x10⁰     (b) –7x10¹
   (c) –6x10^-3   (d) 5x10⁶

CALCULATING WITH SCIENTIFIC NOTATION NUMBERS Here are the rules for multiplying, dividing, adding and subtracting without a calculator.

• To multiply two scientific notation numbers, multiply the coefficients and add the exponents.
                       EXAMPLE: (3×10¹)(2.5×10^-2) = 7.5×10^-1
• To divide two scientific notation numbers, divide the coefficients and subtract the exponents.
                       EXAMPLE: (9×10²⁰)/(3×10¹⁵) = 3×10⁵
• Addition and subtraction are harder – you can only add or subtract numbers with the same exponent. Adjust one of the numbers so it has the same exponent as the other, and only then can you add or subtract the coefficients.
                       EXAMPLE: Let's say you want to add two time intervals, 2×10⁴seconds and 3x10³seconds.
                            Adjust the first number so it has the same exponent as the second number, then add.
                                      (2×10⁴)+(3×10³) = (20×10³)+(3×10³) = 23×10³ = 2.3×10⁴
                                    The last step we did so the coefficient would be between 1 and 10.

What is the reasoning behind these rules? Here are two short videos that explain:

• multiplication & division
• addition and subtraction

Do the following problems purely in scientific notation, without using your calculator.

10. Do these multiplication problems without your calculator.
   (a) (2x10²)(3x10³)
   (b) (4.5x10¹)(2x10⁰)
   (c) (7x10⁹)(–3x10¹)
   (d) (7x10⁵)(1x10²)
               Solution video

11. Do these division problems without your calculator.
   (a) (2x10²)/(3x10³)
   (b) (4.5x10^-1)/(2x10⁰)
   (c) (7x10⁹)/(-3x10¹)
   (d) (7x10^-5)/(1x10^-2)
              Solution video

12. Do these addition problems without your calculator.
   (a) (2x10²)+(3x10³)
   (b) (4.5x10^-1)+(2x10⁰)
   (c) (7x10⁹)+(-3x10¹)
   (d) (7x10^-5)+(1x10^-2)
              Solution video

13. Do these subtraction problems without your calculator.
   (a) (2x10²)–(3x10³)
   (b) (4.5x10^-1)–(2x10⁰)
   (c) (7x10⁴)–(–3x10¹)
              Solution video

Using Scientific Notation on your Calculator  If you are using a calculator, things are a little easier, but you still need to know what you're doing. All scientific calculators (which you must have for this course) have a special button to make it easier to use scientific notation. On most of them, it is labeled EE or EXP.

It’s impossible to give directions for every calculator on the market, but I will mention one of the more popular brands, Texas Instruments. On most of the Texas Instruments graphing calculator models, the EE is above the comma key: you have to hit the 2nd function key to access it. To type in the number 2×10⁴, the keys you would hit are
                           
YOU DON'T TYPE IN THE TIMES KEY, NOR THE NUMBER 10! The screen will display 2E4, where the E stands for “times ten to the power of”. This saves space on the calculator display. After you hit enter, the screen will either show 20000 or 2E4, depending on the display mode you have set. To change the mode, hit the MODE key and select Sci if you want all results displayed in scientific notation, or Normal if you don’t. In Normal mode the calculator will display scientific notation only for very large and very small numbers.

To do the whole addition problem shown above, here is the sequence of keys you would hit:
                         

On the TI–85, TI–86 and TI–89, the EE key is its own dedicated key, so you don’t need to hit the 2nd function key to input scientific notation numbers.

If you need help learning to use scientific notation on your particular calculator, please ask.

Using an "E" or "e" to stand for "times ten to the power of" is also used in computer programming languages and spreadsheets (such as Microsoft Excel).

Do these practice problems using your calculator.

14. (7.4x10⁵⁵)x(4.32x10^-11)
15. (3x10⁸)²
16. (3.33x10⁸⁴)/(3x10⁰)

17. Check your answers to 10-13 using your calculator.

Additional Activities & Practice

18. Let's say you are explaining math to your little sibling. In complete sentences, how would you explain the distinction between the three numbers 3⁸, 3×10⁸ and 3E8?

19. How many electrons would it take to equal the mass of Earth?

20. There are about 125 billion galaxies in the Universe, acording to a recent NASA estimate based on Hubble Space Telescope data. If galaxies have 200 billion stars on average (about the number in our galaxy, the Milky Way) how many stars are there in the Universe?