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:
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:
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
1. The mass of an electron is 9.11x10-31kg.
Fill in each blank with the word POSITIVE or NEGATIVE.
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:
• To multiply two scientific notation numbers, multiply the coefficients
and add the exponents.
What is the reasoning behind these rules? Here are two short videos that explain:
11. Do these division problems without your calculator.
12. Do these addition problems without your calculator.
13. Do these subtraction problems without your calculator.
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
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.
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?