{Scientific Notation

Measuring the very largest distances and the very smallest portions…

In order to understand “scientific notation” we first need to understand exponents. Most of us understand the concept – because we know the numbers 1 – 12 “squared.”

12 = 1 22 = 4 32 = 9

42 = 52 = 62 =

72 = 82 = 92 = 81

102 = 100 112 = 122 =Understanding Exponents

We can figure these in reverse as well! This is taking the square root of a number!

√144 = ____ √121 = ____ √81 = ____

√100 = ____ √1 = ____ √4 = ____

√9 = ____ √25 = ____ √36 = ____

√16 = ____ √64 = ____ √49 = ____

Square Roots

{

You should know by now that a number “squared” is just a number being multiplied by itself.

Consider the equation: 22 = 4

We know that this is just another way to state:

2 x 2 = 4.

So what about this slightly different equation:?

base exponent

23

Other Exponential Forms

So what is the value of 23?

Exponent

Base 23

{ { Two to the third power =

23 =

2 x 2 x 2 =

8Two cubed equals

eight.

What other examples can we solve involving cubed numbers?

A. 13 = 1 x 1 x 1 = _____ E. 63 = 6 x 6 x 6 = _____

B. 33 = 3 x 3 x 3 = _____ F. 73 = 7 x 7 x 7 = _____

C. 43 = 4 x 4 x 4 = _____ G. 83 = 8 x 8 x 8 = _____

D. 53 = 5 x 5 x 5 = _____ H. 103 = 10 x 10 x 10 =

…and we can keep this up all day!

Any number to the power of zero is equal to one.

This is true no matter how larger or small a number is – and no matter whether the number is positive or negative.

Example 1. 5870 = 1Example 2. 550 = 1Example 3. 10 = 1

The only exception to the rule would be 00, because zero to the zero power is undefined. It doesn’t exist.

Special Exponents: n0

Any number to the power of one is equal to the number.

This is true no matter how larger or small a number is – and no matter whether the number is positive or negative.

Example 1. 5871 = 587Example 2. 551 = 55Example 3. 11 = 1

Special Exponents: n1

Solve these equations with exponents:

A. 23430 = _____ B. 3451 = _____

C. -2580 = _____ D. -8541 = _____

E. 2651 = _____ F. 01 = _____

G. 53 = _____ H. 24 = _____

Practice with exponents….

The number ten is a very important one in mathematics – and for many reasons. Our system of counting is a “base ten” system. Meaning that the concept of “place value” in our counting system is achieved by advancing in units of ten.

For example , ten units of one = 10. And ten units of ten = 100.Ten units of 100 = 1000; ten units of 1000 = 10, 000. And so on, infinitely!

This is place value. And you understand it, right?10 to Exponential Powers, or 10n

Here are some very simple examples. Can you determine the proper relationship?

A. 7 <, >, or = 10 B. 45 <, >, or = 100C. 659 <, >, or =1,000 D. 10,000 <, >, or = 1, 000 E. 89,899 <, >, or = 100,000

Examples of place value.

You know that 7 is less than 10, even though the

number seven is larger than both 1 an 0 – or even

1 and 0 combined.

You know that 10,000 is greater than 1,000 even

though the numbers involved are essentially the

same.

And you know that 89, 899 is still less than

100,000 – even though every number in 89, 899 is

larger than the 1 and zeroes in 100,000!We know about place value!

Scientific notation is simply another way to measure place value. We use scientific notation in two basic contexts.

1. When we are using extremely large numbers!

OR 2. When we are using infinitesimally

small numbers!

Scientific Notation

{ Consider this example: What is the distance from the Earth to the Sun in miles?

The answer is approximately 93 Million miles!

We can write this out longhand – 93, 000, 000 miles.

Or, we can abbreviate the number using scientific notion.The Distance from the Earth to the Sun!

{The Earth is 93,000,000 miles from the Sun.

9.3 X 107 miles from the Sun.

Because our system of place value is base ten, we can easily measure large numbers – and smaller numbers, too – by using our knowledge of the number ten’s exponential values!

CHECK IT!

100 = 1 105 = 100000101 = 10 106 = 1000000102 = 100 107 = 10000000103 = 1000 108 = 100000000104 = 10000 109 = 1000000000Ten to the nth power! And we

can do this for any power of 10… Infinitely!

Consider these examples:

A. The distance between the Sun and the planet Jupiter : 483, 700, 000 miles.

B. The number of people on the planet Earth. Total population: 6, 960, 000, 000.

Representing large numbers in Scientific Notation.

Since we all know the value of 10n, we are able to use exponents of ten to represent the place value of large numbers.

The distance between the sun and the planet Jupiter, then, becomes this multiplication product: 4.837 x 108.

We know the value of 108 is 100, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication problem.

4.837 x 100, 000, 000 = 483, 700, 000.

Numbers in Scientific Notation.

Since we all know the value of 10n, we are able to use exponents of ten to represent the place value of large numbers.

The population of the planet Earth, approximately 6.96 billion people, or 6, 960, 000, 000 becomes this multiplication product: 6.96 x 109.

We know the value of 109 is 1, 000, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication problem.

6.96 x 1,000, 000, 000 = 6, 960, 000, 000 or 6.96 Billion!

The World Population in Scientific Notation.

A shorter method of writing numbers in scientific notation is to identify the exponent of 10 in the number and literally move the decimal by that number of “places.”

Consider these examples. Note that the .0 at the end of each number does not change it’s value at all! 1.0 = 1, right?

A. 1.000 x 103 = 1, 000.0B. 7.55 x 106 = 7, 550, 000.0C. 3.65 x 1021 = 3, 650, 000, 000, 000, 000, 000,

000.0

When we multiply by ten…

Write each of the numbers below in Scientific Notation:

A. 7,000, 000, 000 B. 8, 500, 000 C. 5,000 D. 25, 000, 000, 000 E. 63, 000, 000 F. 9, 600

G. The United States of America’s current

national debt: $14, 700, 000, 000, 000.

(Yes, you need to use scientific notation for that!)

When we represent the number in scientific notation, we indicate how small the number is by using negative exponents. A negative exponent indicates the number of times a number is divided by itself!

Here are some simple examples:

2-1 = ½ 2-2 = ¼ 4-1 = ¼ 4-2 = ⅟16

5-1 = ⅟5 5-3 = ⅟125

Examples of scientific notation for very, very small values.

When we use negative exponents of the number 10, we see a pattern emerge – similar to the positive exponents.

A1. 101= 10 A2. 10-1 = ⅟10

B1. 102 = 100 B2. 10-2 = ⅟100 C1. 103 = 1000 C2. 10-3 = ⅟1000

D1. 107 = 10000000 D2. 10-7 = ⅟10000000

What do we call the relationships here? It is the same as the relationship between the number 2 and the number ½. Or the number 4 and the number ¼. Negative Exponents of 10 indicate the number of time a number has been divided by 10.

The width of a plant cell is 0.00001276 meters wide.

If we move the decimal place five decimal places to the right, we have changed the number to 1.276.

In order to accurately represent the number in scientific notation, we have to divide it by 105. (Or, multiply by the inverse – 10-5.)

The width of a plant cell can be rewritten as:

1.276 x 10-5 meters wideAn example of scientific notation for very small numbers.

The radius of a hydrogen (H) atom is approximately 2.5 × 10-11 meters.

We can figure out what this number actually signifies by moving the decimal place 11 units of place value in the negative direction! (LEFT!)

The radius of an atom of H is .000000000025 meters.

The radius of an atom of hydrogen.

How quickly does light travel?

The time it takes for light to travel 1 meter is approximately .0000000033 seconds!

How can we write this very small number out in scientific notation?

First, count the number of place value units you have to move the decimal into position – to the right of the first significant figures, the threes!

What is the time it takes for light to travel one (1) meter (m) in seconds?

.0000000033 seconds We must first identify the “significant figures” in our number. In this case, the numbers “3” and “3” are our “sig figs.”

Now, we move the decimal to the right – in this case, by 9 “place values.”

In order to properly represent our new number, 3.3, we will have to divide it by 109. We write this out in scientific notation not by dividing, but by multiplying by the number’s inverse.

The time it takes for light to move 1 meter is 3.3 x 10-9 seconds.

The wavelength of green light is approximately 5.60 x 10-7 meters.

What does this number mean? It means that the wavelength of green

light is 5.60 meters divided by 10,000,000 – a very small height indeed!

To see the number as a decimal, we simply move the decimal seven place value units to the left – the negative direction on any number line.

The height of green light waves?.00000056 meters long.

Rewrite each of these very, very small numbers in scientific notation:

A. .000000000536 metersB. .00016 kilogramsC. .0000000555 seconds Now, write out these very, very small numbers as

decimals.A. 5.36 x 10-6

B. 7.12 x 10-15 C. 3.68 x 10-4Practice makes perfect!

Write out each of the following numbers:

1. Avogadro’s number: 6.022 x 1023 2. Meters traveled per second by light: 2.99 X 106

3. Gross Domestic Product of the United States, 2010: 14.7 x 1012

Now, write each of these numbers in scientific notation:

4. The population of China: 1, 300, 000, 000 people.5. Estimated wealth of Bill Gates, owner of Microsoft: $56,000,000,000 6. Gallons of water in the Pacific Ocean: 187,189,915,062,857,142,857 gallons.

Practice makes perfect!