What is scientific Notation?

Scientific notation is a way of expressing really big numbers or really small numbers.

It is most often used in “scientific” calculations where the analysis must be very precise.

IT IS AN ABBREVIATION!

Scientific Notation

A number is expressed in scientific notation when it is in the form

a x 10n

where a is between 1 and 10 and n is an integer FOR EXAMPLE: 2.898 x 108

When using Scientific Notation, there are two kinds of exponents: positive and negative--

Positive Exponent: REALLY BIG NUMBERS

2.35 x 108

Negative Exponent: REALLY SMALL NUMBERS

3.97 x 10-7

When using Scientific Notation, there are two kinds of exponents: positive and negative--

Positive Exponent: REALLY BIG NUMBERS

2.35 x 108

Negative Exponent: REALLY SMALL NUMBERS

3.97 x 10-7

Converting from standard notation to scientific notation.

210,000,000,000,000,000,000,000 Where is the decimal point now?

Where would you put the decimal to make this number be between 1 and 10?

*Between the 2 and the 1So, 2.1 becomes the coefficient.

And, because the decimal moved 23 places, the answer is 2.1 x 10²³

*The exponent is positive because our original number was greater than one.

Example

• Given: 289,800,000

• Where would you put the decimal to make the number be between 1 and 10?

• Answer: 2.898 x 10⁸ *The exponent is positive because our original

number was greater than one.

Converting from standard notation to scientific notation.

Example

• Given: 0.000567

• Where would you put the decimal to make the number be between 1 and 10?

• Answer: 5.67 x 10⁻⁴ *The exponent is negative because our original number was less than one.

Converting from standard notation to scientific notation.

When changing from scientific notation to standard notation, the exponent tells how many spaces to move the decimal, and in what direction:

4.08 x 103 = 4 0 8 In this problem, the exponent is +3, so the decimal moves 3 spaces to the right.Don’t forget to fill in your zeroes!

Answer: 4,080

When changing scientific notation to standard notation, the exponent tells you if you should move the decimal:

With a negative exponent, move the decimal to the left:4.08 x 10-3 =

Don’t forget to fill in your zeroes!

Answer 0.00408

An easy way to remember this is:

• If an exponent is positive, the number gets larger, so move the decimal to the right.

• If an exponent is negative, the number gets smaller, so move the decimal to the left.

Try changing these numbers to and from Standard Notation to Scientific Notation:

1) .00008376

2) 56730000

3) 3.45 x 10-5

4) 9.872 x 106

Comparing Numbers in Scientific Notation

Which number is smaller? 1.23 x 1012 or 8.75 x 1010

Which number is greater? 6.25 X 10-6 or 7.43 x 10⁻⁴

*Remember, negative numbers are the opposite from positive numbers. So, -6 is smaller than -4!

What is the Pattern? 32000 x 10¯³ = 32

3200 x 10¯² = 32 320 x 10¯¹ = 32 32 x 10° = 32 3.2 x 10¹ = 32 .32 x 10² = 32 .032 x 10³ = 32

Notice how as the coefficient decreases, and the exponent increases, but the answer is always 32!

Even though only one these numbers is correctly written in Scientific Notation, it is necessary to be able to manipulate our decimal point for using operations with Scientific Notation.

Problem: (2.8 x 103)(5.1 x 10-7)1. multiply the coefficient 2. Keep your base and add the exponents3. You will get: 14.28 x 10-4

4. To put the answer in scientific notation, you have to move the decimal one place LEFT.

Answer: 1.428 x 10-3

Multiplying with Scientific Notation- Use multiplication rules and laws of exponents to evaluate:

(3 x 103)(8 x 109)1. Step one? 2. Step two? 3. Step three? 4. Are we done?

Let’s Try One!

4.5 x 105

1.5 x 102

1. Divide the coefficient 2. Keep your base and subtract the exponents

Write your answer in scientific notation. 4.5 = 3.00 105 = 103 1.5 102

3.00 x 103

Dividing with Scientific Notation-Use division rules and laws of exponents to evaluate:

7.2 x 10-3

1.2 x 102

1. Step one? 2. Step two? 3. Step three? 4. Are we done?

Which is scientifically notated as 6 x 10 -5

The answer in decimal notation is 0.00006

Let’s try one!

What is the Pattern? 32000 x 10¯³ = 32

3200 x 10¯² = 32 320 x 10¯¹ = 32 32 x 10° = 32 3.2 x 10¹ = 32 .32 x 10² = 32 .032 x 10³ = 32

Notice how as the coefficient decreases, and the exponent increases, but the answer is always 32!

Even though only one these numbers is correctly written in Scientific Notation, it is necessary to be able to manipulate our decimal point for using operations with Scientific Notation.

Adding and Subtracting with Scientific Notation

• When adding or subtracting numbers in scientific notation, the exponents must be the same.

• If they are different, you must convert one of the numbers so that they will have the same exponent.

Adding With the Same Exponent

• (3.45 x 103) + (6.11 x 103)• 3.45 + 6.11 = 9.56• 9.56 x 103

Adding With Different Exponents

• (4.12 x 106) + (3.94 x 104)

• (412 x 104) + (3.94 x 104)• 412 + 3.94 = 415.94• 415.94 x 104

• Express in proper form: 4.15 x 106

To make the exponents the same, manipulate your number by moving your decimal and changing your exponent by equal amounts (levels).

Example 22.46 X 106 + 3.4 X 103

246000 + 3400 = 249400

2.494 x 10⁵

You can also expand your numbers into standard form and then convert them back into Scientific Notation.

Put your answer in Scientific Notation!

Subtracting With the Same Exponent

• (8.96 x 107) – (3.41 x 107)• 8.96 – 3.41 = 5.55• 5.55 x 107

Subtracting With Different Exponents

• (4.23 x 103) – (9.56 x 102)

• (42.3 x 102) – (9.56 x 102)• 42.3 – 9.56 = 32.74• 32.74 x 102

• Express in proper form: 3.27 x 103

To make the exponents the same, manipulate your number by moving your decimal and changing your exponent by equal amounts (levels).

Example 25.762 X 103 – 2.61 X 102

5762 – 261 = 5501

• Answer: 5.501 X 103

You can also expand your numbers into standard form and then convert them back into Scientific Notation.

Put your answer in Scientific Notation!