Standard Standard FormForm

(also referred to as "scientific notation“)

In science, we deal with some In science, we deal with some very very LARGELARGE numbers: numbers:

1 mole = 6020000000000000000000001 mole = 602000000000000000000000

In science, we deal with some In science, we deal with some very very SMALLSMALL numbers: numbers:

Mass of an electron =Mass of an electron =0.000000000000000000000000000000091 kg0.000000000000000000000000000000091 kg

Standard FormStandard Form

Imagine the difficulty of Imagine the difficulty of calculating the mass of 1 mole calculating the mass of 1 mole of electrons!of electrons!

0.00000000000000000000000000000000.000000000000000000000000000000091 kg91 kg x 602000000000000000000000x 602000000000000000000000

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Standard Form:Standard Form:

A method of representing very large A method of representing very large or very small numbers in the form:or very small numbers in the form:

M x 10M x 10nn

MM is a number betweenis a number between 11 andand 1010

nn is an integeris an integer

For very large numbers and very small For very large numbers and very small numbers, standard form is numbers, standard form is more more conciseconcise..

2 500 000 000

Step 1: Insert an understood decimal pointStep 1: Insert an understood decimal point

.

Step 2: Decide where the decimal must Step 2: Decide where the decimal must end end up so that one number is to its up so that one number is to its leftleftStep 3: Count how many places you Step 3: Count how many places you bounce bounce the decimal pointthe decimal point

123456789

Step 4: Re-write in the form Step 4: Re-write in the form M x 10M x 10nn

2.5 x 102.5 x 1099

The exponent is the number of places we moved the decimal.

0.00005790.0000579

Step 2: Decide where the decimal must Step 2: Decide where the decimal must end end up so that one number is to its up so that one number is to its leftleftStep 3: Count how many places you Step 3: Count how many places you bounce bounce the decimal pointthe decimal pointStep 4: Re-write in the form M x 10Step 4: Re-write in the form M x 10nn

1 2 3 4 5

5.79 x 105.79 x 10-5-5

The exponent is negative because the number we started with was less than 1.

Timberlake lecture plus 9

More ExamplesMore Examples

Given: 289,800,000

Use: 2.898 (moved 8 places to the left)

Answer: 2.898 x 108

Given: 0.000567

Use: 5.67 (moved 4 places to the right)

Answer: 5.67 x 10–4

Timberlake lecture plus 10

Learning CheckLearning Check

Express these numbers in standard form:

1) 405789

2) 0.003872

3) 3000000000

4) 2

5) 0.478260

Timberlake lecture plus 11

SolutionSolution

Express these numbers in standard form:

1) 4.05789 x 105

2) 3.872 x 10–3

3) 3 x 109

4) 2 x 100

5) 4.7826 x 10–1

Move the decimal place to the Move the decimal place to the right for a positive exponent 10.right for a positive exponent 10.

Move the decimal place to the Move the decimal place to the left for a negative exponent 10.left for a negative exponent 10.

Use zeros to fill in places.Use zeros to fill in places.

Coming out of Standard FormComing out of Standard Form

Timberlake lecture plus 13

ExamplesExamples

Given: 5.093 x 106

Use: 5,093,000 (moved 6 places to the right)

Given: 1.976 x 10–4

Use: 0.0001976 (moved 4 places to the left)

PERFORMING PERFORMING CALCULATIONS CALCULATIONS IN STANDARD IN STANDARD

FORMFORM

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

ReviewReview::Standard form expresses a Standard form expresses a number in the form:number in the form:

M x 10M x 10nn

1 1 M M 1010

n is an n is an integerinteger

4 x 104 x 1066

+ 3 x 10+ 3 x 1066

IFIF the exponents the exponents are the same, we are the same, we simply add or simply add or subtract the subtract the numbers in front numbers in front and bring the and bring the exponent down exponent down unchanged.unchanged.

77 x 10x 1066

4 x 104 x 1066

- 3 x 10- 3 x 1066

The same holds The same holds true for true for subtraction in subtraction in standard form.standard form.11 x 10x 1066

4 x 104 x 1066

+ 3 x 10+ 3 x 1055

If the exponents If the exponents are NOT the are NOT the same, we must same, we must move a decimal to move a decimal to makemake them the them the same.same.

4.00 x 104.00 x 1066

+ + 3.00 x 103.00 x 1055 + + .30 x 10.30 x 1066

4.304.30 x 10x 1066

Move the Move the decimal decimal on the on the smallersmaller number!number!

4.00 x 104.00 x 1066

A Problem for A Problem for you…you…

2.37 x 102.37 x 10-6-6

+ 3.48 x 10+ 3.48 x 10-4-4

2.37 x 102.37 x 10-6-6

++ 3.48 x 103.48 x 10-4-4

Solution…Solution…002.37 x 10002.37 x 10--

66

+ 3.48 x 10+ 3.48 x 10-4-4

Solution…Solution…0.0237 x 100.0237 x 10-4-4

3.5037 x 103.5037 x 10-4-4