Scientific Notation

 

Learning Objective(s)

·         Convert between scientific and decimal notation.

·         Multiply and divide using scientific notation.

·         Solve application problems.

 

 

When working with very large or very small numbers, scientists, mathematicians, and engineers often use scientific notation to express those quantities. Scientific notation uses exponential notation. The following are examples of scientific notation.

 

Light year: number of miles light travels in one year, about 5,880,000,000,000

            Scientific notation is 5.88 x 10¹² miles.

 

hydrogen atom: has a diameter of about 0.00000005 mm

Scientific notation is 5 x 10^-8 mm

 

Computation with very large numbers is made easier with scientific notation.

 

 

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.

 

Since it’s so useful, let’s look more closely at the details of scientific notation format.

 

Scientific Notation A positive number is written in scientific notation if it is written as a x 10n where the coefficient a has a value such that 1 ≤ a < 10 and n is an integer.

 

Look at the numbers below. Which of the numbers is written in scientific notation?

 

 

 

Which number below is written in scientific notation?   A) 4.25 x 100.08   B) 0.425 x 107   C) 42.5 x 105   D) 4.25 x 106     Show/Hide Answer A) 4.25 x 100.08 Incorrect. The exponent must be an integer and 0.08 is not an integer. The correct answer is 4.25 x 106.   B) 0.425 x 107 Incorrect. This is not in scientific notation because 0.425 is less than 1. The correct answer is 4.25 x 106.   C) 42.5 x 105 Incorrect. This is not in scientific notation because 42.5 is greater than 10. The correct answer is 4.25 x 106.   D) 4.25 x 106 Correct. This is scientific notation. 4.25 is greater than 1 and less than 10, and 6 is an integer.

 

 

Now let’s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.

 

 

 

To write a large number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.

 

Let’s look at an example.

180,000. =      18,000.0  x  10¹

                                                              1,800.00  x 10²

                                                                 180.000 x 10³

                                                                   18.0000 x 10⁴

                                                                     1.80000 x 10⁵

 

                                    180,000 = 1.8 x 10⁵

 

Notice that the decimal point was moved 5 places to the left, and the exponent is 5.

 

 

The world population is estimated to be about 6,800,000,000 people. Which answer expresses this number in scientific notation?   A) 7 x 109   B) 0.68 x 1010   C) 6.8 x 109   D) 68 x 108   Show/Hide Answer A) 7 x 109 Incorrect. Scientific notation rewrites numbers, it doesn’t round them. The correct answer is 6.8 x 109.   B) 0.68 x 1010 Incorrect. Although 0.68 x 1010 is equivalent to 6,800,000,000. 0.68 is not the form for scientific notation as 0.68 is not a number between 1 and 10. The correct answer is 6.8 x 109.   C) 6.8 x 109 Correct. The number 6.8 x 109 is equivalent to 6,800,000,000 and uses the proper format for each factor.   D) 68 x 108 Incorrect. Although 68 x 108 is equivalent to 6,800,000,000, it is not written in scientific notation as 68 is not between 1 and 10. The correct answer is 6.8 x 109.

 

 

Advanced Question Represent 1.00357 x 10-6 in decimal form.   A) 1.00357000000 B) 0.000100357 C) 0.000001357 D) 0.00000100357   Show/Hide Answer A) 1.00357000000 Incorrect. You added six 0s to the end of the decimal. This does not change its value at all; 1.00357000000 = 1.00357, which is not equal to 1.00357 x 10-6. The correct answer is 0.00000100357.                                                                                                                       B) 0.000100357 Incorrect. You moved the decimal point in the correct direction, but you did not move it enough places. The correct answer is 0.00000100357.   C) 0.000001357 Incorrect. You moved the decimal point the correct number of spaces, but the number you created is different than the number you started with: 1.00357 x 10-6 ≠ 0.000001357. Remember that the zeroes in between 1 and 3 must also be included in the final number. The correct answer is 0.00000100357.   D) 0.00000100357 Correct. The exponent is -6. You moved the decimal point 6 spots to the left, creating the decimal 0.00000100357.

 

To write a small number (between 0 and 1) in scientific notation, you move the decimal to the right and the exponent will have to be negative.

 

                              0.00004     =        00.0004   x  10^-1

                                                000.004   x  10^-2

                                                            0000.04   x   10^-3

                                                            00000.4   x  10^-4

                                                            000004.  x  10^-5

 

                                        0.00004     =   4 x 10^-5

 

You may notice that the decimal point was moved five places to the right until you got the number 4, which is between 1 and 10. The exponent is −5.

 

 

You can also write scientific notation as decimal notation. For example, the number of miles that light travels in a year is 5.88 x 10¹², and a hydrogen atom has a diameter of 5 x 10^-8 mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

 

 

For each power of 10, you move the decimal point one place. Be careful here and don’t get carried away with the zeros—the number of zeros after the decimal point will always be 1 less than the exponent because it takes one power of 10 to shift that first number to the left of the decimal.

 

 

Rewrite 1.57 x 10-10 in decimal notation.   A) 15,700,000,000   B) 0.000000000157   C) 0.0000000000157   D) 157 x 10-12     Show/Hide Answer A) 15,700,000,000 Incorrect. You moved the decimal point in the wrong direction. The exponent is negative, so to convert to decimal format move the decimal point to the left, not the right. The correct answer is 0.000000000157.   B) 0.000000000157 Correct. The expression has a negative exponent, so you move the decimal point 10 places to the left to convert it to decimal notation, 0.000000000157.   C) 0.0000000000157 Incorrect. You inserted 10 zeros between the number and the decimal point. Move the decimal point 10 places to the left instead. The correct answer is 0.000000000157.   D) 157 x 10-12 Incorrect. This number is equivalent to the original number, but it is not in decimal notation. The correct answer is 0.000000000157.

 

 

 

Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of the properties of numbers and the rules of exponents that you may recall. To multiply numbers in scientific notation, first multiply the numbers that aren’t powers of 10 (the a in a x 10ⁿ). Then multiply the powers of ten by adding the exponents.

 

This will produce a new number times a different power of 10. All you have to do is check to make sure this new value is in scientific notation. If it isn’t, you convert it.

 

Let’s look at some examples.

 

 

 

 

 

 

In order to divide numbers in scientific notation, you once again apply the properties of numbers and the rules of exponents. You begin by dividing the numbers that aren’t powers of 10 (the a in a x 10ⁿ). Then you divide the powers of ten by subtracting the exponents.

 

This will produce a new number times a different power of 10. If it isn’t already in scientific notation, you convert it, and then you’re done.

 

Let’s look at some examples.

 

 

 

 

 

 

Notice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator.

 

 

Evaluate (4 x 10-10)(3 x 105) and express the result in scientific notation.   A) 1.2 x 10-4   B) 12 x 10-5   C) 7 x 10-5   D) 1.2 x 10-50     Show/Hide Answer A) 1.2 x 10-4 Correct. 1.2 x 10-4 is an accurate computation and correct scientific notation.   B) 12 x 10-5 Incorrect. Almost correct, but now you have to convert the coefficient 12 into scientific notation. 12 is greater than 10 and scientific notation requires this number to be greater than or equal to 1 and less than 10. The correct answer is 1.2 x 10-4.   C) 7 x 10-5 Incorrect. Multiply, not add, the numbers 4 and 3. The correct answer is 1.2 x 10-4.   D) 1.2 x 10-50 Incorrect. Add, not multiply, exponents. The correct answer is 1.2 x 10-4.

 

 

Advanced Question Evaluate (3.15 x 104)(5.15 x 10-7) and express the result as a decimal.   A) 0.0162225 B) 162225 C) 0.000162225 D) 16.2225   Show/Hide Answer A) 0.0162225 Correct. 3.15 x 5.15 = 16.2225, and 104 x 10-7 = 10-3. The exponent is negative, so to convert to decimal format move the decimal point three spaces to the left for a value of 0.0162225.   B) 162225 Incorrect. The expression has a negative exponent, so you move the decimal point 3 places to the left, not to the right, to convert it to decimal notation. The correct answer is 0.0162225.   C) 0.000162225 Incorrect. You moved the decimal point in the correct direction, but not the correct number of spots. Remember that 104 x 10-7 = 10-3, so you have to move the decimal point three spots to the left. The correct answer is 0.0162225.   D) 16.2225 Incorrect. 3.15 x 5.15 = 16.2225, but it looks like you forgot to multiply 104 x 10-7. The correct answer is 0.0162225.

 

 

 

Scientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of 10. The format is written a x 10ⁿ, where 1 ≤ a < 10 and n is an integer.

 

To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.