Scientific Notation

 

Learning Objective(s)

·         Describe the rules of scientific notation.

·         Convert numbers between decimal and scientific notation.

 

Introduction

 

When working with very large or very small numbers, scientists, mathematicians, and engineers use scientific notation to express those quantities. Scientific notation is mathematical shorthand, based on the idea that it is easier to read one exponent than it is to count many zeroes in a number. Very small or very large numbers use less space when written in scientific notation because place values are expressed as a power of 10. Computation with long numbers is made easier with scientific notation.

 

Learning to Use Scientific Notation

 

A human red blood cell is very small and is estimated to have a diameter of 0.0065 millimeters. On the other hand, a light year is a very large unit of distance that measures just under 10,000,000,000,000,000 meters. Those figures are both awkward to write, and it would be easy to add or lose a zero or two along the way. But in scientific notation, a red blood cell diameter is written as 6.5 x 10-3 millimeters, and a light year as roughly 1 x 1016 meters. Those figures are easier to use than the longhand versions.

 

Notice that it is the exponent that tells us if the term is a very large or a very small number. If a number is ≥ 1 in standard decimal notation, the exponent will be ≥ 0 in scientific notation. In other words, large numbers require positive powers of 10.

 

If a number is between 0 and 1 in standard notation, the exponent will be < 0 in scientific notation.  Small numbers are described by negative powers of 10.

 

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

 

Scientific Notation Format

 

The general form for a number in scientific notation is a x 10n where  and n is an integer.

 

 

We must pay close attention to these conventions in order to write scientific notation correctly. Let’s look at some examples:

 

 

Number

Scientific Notation?

Explanation

1.85 x 10-2

yes

 

-2 is an integer

no

 is not an integer

0.82 x 1014

no

0.82 is not ≥ 1

10 x 103

no

10 is not < 10

 

Only numbers that follow the proper conventions for all parts of the expression are considered to be in true scientific notation.

 

Which number below is written in proper scientific notation format?

                    

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) Incorrect. The exponent must be an integer and 0.08 is not an integer. The correct answer is 4.25 x 106.

 

B) Incorrect. This is not in proper form because 0.425 is less than 1. The correct answer is 4.25 x 106.

 

C) Incorrect. This is not in proper form because 42.5 is greater than 10. The correct answer is 4.25 x 106.

 

D) Correct. This reflects the proper format because 4.25 is greater than 1 and less than 10 and 6 is an integer.

 

 

Moving From Decimal to Scientific Notation

 

Now that we understand the format of scientific notation, 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 table below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.

 

Large Numbers

Small Numbers

Decimal Notation

Scientific Notation

Decimal Notation

Scientific Notation

500.0

5 x 102

0.05

5 x 10-2

80,000.0

8 x 104

0.0008

8 x 10-4

43,000,000.0

4.3 x 107

0.00000043

4.3 x 10-7

62,500,000,000.0

6.25 x 1010

0.000000000625

6.25 x 10-10

 

 

Let’s start with large numbers. To write a large number in scientific notation, we first have to move the decimal point to a number between 1 and 10. Since moving the decimal point changes the value, we have to apply multiplication by the power of 10 that will yield an equivalent value to the original. To figure out the exponent, we just count the number of places we moved the decimal sideways. That number is the exponent for the power of 10.

 

Let’s look at an example. To rewrite 180,000 in scientific notation, we first move the decimal point to the left until we have a number greater than or equal to 1 and less than 10. The decimal point is not written in 180,000, but if it were it would be after the last zero.  If we start moving the decimal sideways one place at a time, we’ll get to 1.8 after 5  shifts:

180000.

18000.0

1800.00

180.000

18.0000

1.80000

So now we know both the number (1.8) and the exponent for the power of 10 multiplier that preserves the original value (5).  In .scientific notation 180,000 is written as 1.8 x 105.

 

The world population is estimated to be about 6,800,000,000 people. Which answer below correctly 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) Incorrect. Scientific notation rewrites numbers, it doesn’t round them off. The correct answer is C) 6.8 x 109.

 

B) Incorrect. Although 0.68 x 1010 is equivalent to 6,800,000,000, 0.68 is not the proper format. It is not greater than or equal to 1, or less than 10, a requirement for correct scientific notation. The correct answer is C) 6.8 x 109.

 

C) Correct. This number is equivalent to 6,800,000,000 and used the proper format for each factor.

 

D) Incorrect. Although 68 x 108 is equivalent to 6,800,000,000, it is not the proper format Because 68 is not greater than or equal to 1, or less than 10. The correct answer is C) 6.8 x 109.

 

 

The process for moving between decimal and scientific notation is the same for small numbers (between 0 and 1), but in this case the decimal moves to the right, and the exponent will be negative. Consider the small number 0.0004:

 

0.0004

00.004

000.04

0000.4

00004.

 

We moved the decimal point sideways until we got the number 4, which is between 1 and 10 as required. It took 4 moves, but they were moves that made the number bigger than the original. So we’ll have to multiply by a negative power of 10 to bring the new number back down to the equivalent of the original value. In .scientific notation 0.0004 is written as 4.0 x 10-4.

 

Moving From Scientific to Decimal Notation

 

We can also go the other way—numbers written in scientific notation can be translated into decimal notation.  For example, a hydrogen atom has a diameter of 5 x 10-8 mm. To write this number in decimal notation, we turn that power of 10 back into a series of zeros between the number and the decimal point. Because the exponent is negative, all those zeroes go to the left of the number 5:

 

5 x 10-8

 

5.

0.5

0.05

0.005

0.0005

0.00005

0.000005

0.0000005

0.00000005

 

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

 

 

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) Incorrect. You moved the decimal point in the wrong direction. The exponent is negative, so to convert to decimal format you must move the decimal point to the left, not the right. The correct answer is B) 0.000000000157.

 

B) Correct. The number had a negative exponent, so you moved the decimal point 10 places to the left to convert it to decimal notation.

 

C) Incorrect. You inserted 10 zeroes between the number and the decimal point. You should have moved the decimal point 10 places to the left instead. The correct answer is B) 0.000000000157.

 

D) Incorrect. This number is equivalent to a the original number, but it is not in decimal notation. The correct answer is B) 0.000000000157.

 

 

Multiplying and Dividing Numbers Expressed in Scientific Notation

 

Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of a few properties and rules. 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 we have to do is check to make sure this new value is in scientific notation. If it isn’t, we convert it.

 

Let’s look at an example:

 

Example

Problem

 

(3 x 108)(6.8 x 10-13)

 

 

 

(3 • 6.8)(108 x 10-13)

Regroup using the Commutative and Associative Properties

 

 

(20.4)( 108 x 10-13)

 

 

Multiply the numbers

 

20.4 x 10-5

Add the exponents following the rule of exponents

 

2.04 x 101 x 10-5

Convert 20.4 into scientific notation

 

2.04 x 101+(-5)

Add the exponents following the rule of exponents

Answer

2.04 x 10-4

 

 

 

In order to divide numbers in scientific notation, we once again apply the properties of numbers and the rules of exponents.  We begin by dividing the numbers that aren’t powers of 10 (the a in a x 10n). Then we 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, we convert it, and then we’re done.

 

Let’s look at an example:

 

Example

Problem

 

 

 

Regroup using the Associative Property

 

(0.82)

Divide the numbers

 

0.82 x 10-9 – (-3)

 

0.82 x 10-6

Subtract the exponents

 

(8.2 x 10-1) x 10-6

Convert 0.82 into scientific notation

 

8.2 x 10-1+(-6)

Add the exponents

Answer

8.2 x 10-7

 

 

 

 

Notice that when we divide exponential terms, we 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) Correct. This is accurate computation and correct scientific notation.

 

B) Incorrect. Almost, but now you have to convert this number 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 A) 1.2 x 10-4.

 

C) Incorrect. You should have multiplied, not added, the numbers 4 and 3. The correct answer is A) 1.2 x 10-4.

 

D) Incorrect. Add, not multiply, exponents. The correct answer is A) 1.2 x 10-4.

 

 

 

Summary

 

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  and n is an integer.

 

To multiply numbers in scientific notation, add the exponents. To divide, subtract the exponents.