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.” 1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 7 2 = 49 5 2 = 25 8 2 = 64 6 2 = 36 9 2 = 81 10 2 = 100 11 2 = 121 12 2 = 144

Understanding Exponents

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

√144 = ____ √100 = ____ √9 = ____ √16 = ____ √121 = ____ √1 = ____ √25 = ____ √64 = ____

Square Roots

√81 = ____ √4 = ____ √36 = ____ √49 = ____

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

Consider the equation: 2 2 = 4 We know that this is just another way to state: 2 x 2 = 4.

So what about this slightly different equation:?

exponent { base 2 3

Other Exponential Forms

Exponent Base 2 3

So what is the value of 2

3

?

{ Two to the third power = { 2 x 2 x 2 = 2 3 = 8

Two cubed equals eight.

What other examples can we solve involving cubed numbers?

A. 1 3 = 1 x 1 x 1 = _____ E. 6 3 = 6 x 6 x 6 = _____ B. 3 3 C. 4 3 = 3 x 3 x 3 = = 4 x 4 x 4 = _____ _____ D. 5 3 = 5 x 5 x 5 = _____ F. 7 3 = 7 x 7 x 7 = _____ G. 8 3 = 8 x 8 x 8 = _____ H. 10 3 = 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. Example 2. Example 3.

587 0 55 0 = 1 = 1 1 0 = 1 The only exception to the rule would be 0 0 , because zero to the zero power is undefined. It doesn’t exist.

Special Exponents: n

0

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. Example 2. Example 3.

587 1 = 587 55 1 = 55 1 1 = 1

Special Exponents: n

1

Solve these equations with exponents!

A. 2343 0 = _____ C. -258 0 = _____ E. 265 1 = _____ G. 5 3 = _____ B. 345 1 = _____ D. -854 1 = _____ F. 0 1 = _____ H. 2 4 = _____

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 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 10

n

Here are some very simple examples: A. 7 B. 45 C. 659 D. 10,000 E. 89,899 <, >, or = <, >, or = <, >, or = <, >, or = <, >, or = 10 100 1,000 1, 000 100,000

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 10

7

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!

10 0 10 1 10 2 10 3 10 4 = = = = = 1 10 100 1000 10000 10 10 10 7 10 8 10 5 6 9 = 100000 = 1000000 = 10000000 = 100000000 = 1000000000

Ten to the n

th

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 10 large numbers.

n

, we are able to use exponents of ten to represent the place value of The distance between the sun and the planet Jupiter, then, becomes this multiplication product: 4.837 x 10 8 . We know the value of 10 problem.

8 is 100, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication 4.837 x 100, 000, 000 = 483, 700, 000.

Numbers in Scientific Notation.

    Since we all know the value of 10 large numbers.

n

, we are able to use exponents of ten to represent the place value of The population of the planet Earth, approximately 6.96 billion people, or 6, 960, 000, 000 becomes this multiplication product: 6.96 x 10 9 . We know the value of 10 problem.

9 is 1, 000, 000, 000. And the “significant figures” or “sig figs” in the expression are used to create a “shorthand” multiplication 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 10 3 = 1, 000 .0

B. 7.55 x 10 6 = 7, 550, 000 .0

C. 3.65 x 10 21 = 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 C. 5,000 E. 63, 000, 000 B. 8, 500, 000 D. 25, 000, 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!) Practice? We talkin’ about practice?