The following lesson is one lecture in a series of Chemistry Programs developed by Professor Larry Byrd Department of Chemistry Western Kentucky University

Excellent Assistance has been provided by: Ms. Elizabeth Romero Ms. Kathy Barnes Ms. Padmaja Pakala Ms. Sathya Sanipina

Also known as Exponential Notation Scientific Notation (Part 1) Also known as Exponential Notation

I. Introduction to Exponents Very large number, such as the number of atoms in one gram of uranium (2,530,000,000,000,000,000,000). Very small numbers, such as the distance between two carbon atoms in a molecule (0.000000000154 meters). In the SCIENTIFIC NOTATION METHOD, numbers are expressed as the product of two terms written as follows: A x 10 n 1st part times 2nd part

I. Introduction to Exponents The “First Part” is called A and it is a number that is Equal to “ONE” or the number is “Greater than ONE”, but “Always" less than TEN. The A part may be given as A > 1 or it can be A = 1, but always the A part is less than ten. The above may also be written in Algebra as 1 ≤ A < 10

I. Introduction to Exponents The “Second Part” (10 n) is composed of base ten raised to the value ‘n’. The value ‘n’ is a superscript known as base ten’s exponent ( also called its power ). The exponent (n) may be a zero, or a negative value, or a positive value.

Positive Exponents

Positive Exponents

Any number raised to the zero power is equal to one: 100 = 1 120 = 1 50 = 1 1000 = 1 A0 = 1 Z0 = 1

Negative Exponents Exponents can also be negative (negative exponents), as in the term 10-2. The negative two exponent (-2) indicates that 10 is raised to the positive second power and it is then written in the denominator (bottom) of a fraction and then divided into the number one, which is the numerator of the fraction.

Negative Exponents Thus, 10-2 is equal to one divided by 10+2 : is equal to one divided by 10 times 10, which is equal to one divided by a hundred. As a decimal it is equal to 0.01 (one-hundredth).

Summary of Scientific Notations Exponents BASE-10 Conventional Number 10+4 = 10 x 10 x 10 x 10 = 10,000. 10+3 = 10 x 10 x 10 = 1000. 10+2 = 10 x 10 = 100. 10+1 = 10 = 10. 100 = 1 = 1. Notice: That if the “exponent” is a “positive value” then the conventional number (the-every-day-way-to-write-numbers) is always 10 or it is larger than 10.

Summary of Scientific Notations Exponents 100 = 1 = 1.

Summary of Scientific Notations Exponents 100 = 1. 10-1 = 0.1 10-2 = 0.01 10-3 = 0.001 10-4 = 0.0001 Notice: That if the “exponent” is a “negative value” then the conventional number (the-every-day-way-to-write-numbers) is less than ONE (1.) .

Negative Exponents Remember that a negative exponent will always generate a small number that is always less than one!. Other examples of negative exponents are as follows:

II. Writing in Scientific Notation The number 3698 would be written in Scientific Notation as 3.698 x 103. First, realize that the number 3698 has a decimal point next to the eight (3698.) Next, we will move the decimal point until we get a number that is one or greater than one, but less than ten. First Step: 3698 3 698 . to .

II. Writing in Scientific Notation To show that the decimal point has moved, we will write the number of places the decimal point moved as an exponent (power) of ten. Second Step: 3698. = 3.698 x 10+3 or as 3.698 x 103 = 3.698E3 computer form Remember that if you move the decimal point to the left to produce a number in scientific notation the exponent will always be a positive value.

II. Writing in Scientific Notation Remember that if you move the decimal point to the left to produce a number in scientific notation the exponent will always be a positive value. Examples: 206 = 206. = 2.06 x 10+2 = 2.06 x 102 = computer form = 2.06E2 186000 = 186000. = 1.86 x 10+5 = 1.86 x 105 = computer form = 1.86E5

II. Writing in Scientific Notation Conventional numbers (the every-day-way-of- writing-numbers!) between 1 and 9.99999…etc, have zeroes as their exponents. Example: 9.4 = 9.4 x 100 Numbers equal to or larger than 10 will always have positive exponents!!! Example: 10 = 1.0 x 10+1 ; 94 = 9.4 x 10+1

II. Writing in Scientific Notation In Conventional numbers that are smaller than one, we move the decimal point to the right until we get a number that is one or greater than one, but less than ten (the first part). Example: 0.0333 = 0.0333 = 3.33 x 10-2 = computer form = 3.33E-2 The exponent -2 tells us that we had to move the decimal point two places to the right.

Examples negative exponent. Some other examples: 0.206 = 0.206 = 2.06 x 10-1 = computer form = 2.06E-1 0.000000345 = 0.000000345 = 3.45 x 10-7 = computer form = 3.45E-7 Note, that if the conventional number is less than one, then the number in scientific notation will always have a negative exponent.

Examples More examples of numbers written in scientific notation form: 96,000,000. = 9.6 x 10+7 = 9.6 x 107 = computer form = 9.6E7 0.0046 = 4.6 x 10-3 = 4.6E-3 0.000612 = 6.12 x 10-4 = 6.12E-4

III. Typing in Scientific Notation When you are typing in scientific notation numbers into a computer you must type them in the following way: * 9.6 x 107 = 9.6E7 * 6.12 x 10-4 = 6.12E-4 * 8.64 x 10+2 = 8.64E2 .... etc! When typing numbers in Computer Form (9.6E7) NEVER type in an exponent with its positive (+) sign.

From Scientific Form to Conventional Form IV. If we are given the number in scientific notation form and we are asked to give it in the conventional form, we just reverse the process. Example: 7.19 x 10-3 = 0007.19 x 10-3 = 0007.19 x 10-3 = 0.00719 We just move the decimal point back three places to the left. We must also add zeros when needed.

Examples Some other examples of writing numbers from scientific notation form into conventional form are as follows: 9.6 x 106 = 9.600000 x 10+6 = 9.600000 x 10+6 = 9,600,000 1.23 x 103 = 1.230 x 10+3 = 1.230 x 10+3 = 1230 6.94 x 10-2 = 006.94 x 10-2 = 006.94 x 10-2 = 0.0694

YOU HAVE JUST COMPLETED PART–I OF SCIENTIFIC NOTATION