Scientific Notation- Why?

Also used to maintain the correct number of significant figures.

An alternative way of writing numbers that are very large or very small.

characteristic will be positive Ex: 6.022X1023

602200000000000000000000

Method to express really big or small numbers.

Format is Mantissa x Base Power

We just move the decimal point around.

Decimal part of original number

Decimal you moved

6.02 x 1023

602000000000000000000000

Characteristic

Using the Exponent Keyon a Calculator

EXP

EE

EE or EXP means “times 10 to the…”

How to type out 6.02 x 1023:

6 EE. 0 32 2

6 y x. 0 32 2

x 16 . 0 2 EE 320

y x 32x 16 . 0 2 0

Don’t do it like this…

…or like this…

…or like this:

How to type out 6.02 x 1023:

6 EE. 0 32 2

WRONG!

WRONG!

TOO MUCH WORK.

Example: 1.2 x 105 2.8 x 1013

But instead is written…

=

1 . 2 EE 5

32 . 8 EE 1

Type this calculation in like this:

This is NOT written… 4.3–9

4.2857143 –09Calculator gives…

4.2857143 E–09or…

4.3 x 10–9 4.3 E –9or

Converting Numbers to Scientific Notation

0 . 0 0 0 0 2 2 0 5

1 2 3 4 5

2.205 x 10-5

In scientific notation, a number is separated into two parts.The first part is a number between 1 and 10.

The second part is a power of ten.

Scientific Notation- How

To convert TO scientific notation, move decimal to left or right until you

have a number between 1 & 10. Count # of decimal places moved If original is smaller than 1 than

characteristic will be negative If original is larger than 1 the

If original number is negative, don’t forget to put the – back on the front!

Example:

If you move the decimal to the left the characteristic will be positive

If you move the decimal to the right the characteristic will be negative

Convert 159.0 to scientific notation 1.59 x 102

Convert -0.00634 -6.34 x 10-3

Your Turn

1. 17600.02. 0.00135

3. 10.2

4. -67.30

5. 4.76

6. - 0.1544

7. 301.0

8. -0.000130

1.76 x 104

1.35 x 10-3

1.02 x 101

May drop leading zeros - keep trailing

-6.730 x 101

4.76 x 100

-1.544 x 10-1

3.010 x 102

-1.30 x 10-4

Expand Scientific Notation

If characteristic is positive move decimal to the right

If the characteristic is negative move the decimal to the left

Ex: 8.02 x 10-4

0.000802 -9.77 x 105

-977,000

3-6 10 x 1.23 10 x 4.35

4-6- 10 x 8.7- 10 x 7.5

4-16- 10 x 9.86 10 x 5.76

1111 10 x 3.3 10 x 8.8

= -6.525 x 10-9

= 5.3505 x 103 or 5350.5

= 5.84178499 x 10-13

report -6.5 x 10-9 (2 sig. figs.)

report 5.35 x 103 (3 sig. figs.)

report 5.84 x 10-13 (3 sig. figs.)

= 2.904 x 1023

report 2.9 x 1023 (2 sig. figs.) 8-23 10 x 5.1- 10 x 6.022 = -3.07122 x 1016

report -3.1 x 1016 (2 sig. figs.)

Correcting Scientific Notation The mantissa needs to have one place holder to

the left of the decimal (3.67 not 36.7), look at the absolute value

Count how many decimals places you move and then you will increase or decrease the characteristic accordingly

If you must INCREASE the mantissa, DECREASE the characteristic

If you must DECREASE the mantissa, INCREASE the characteristic

Be careful with negative characteristics! If you decrease 10-3 by two the new value is 10-5

Confused? Example

To correct 955 x 108

Convert 955 to 9.55 – (move decimal left 2 times). Did we increase or decrease 955? 955 is larger than 9.55 so we decreased it -so we

must increase 8 by 2. 955 x 108 becomes 9.55 x 1010

-9445.3 x 10-6

Convert -9445.3 to -9.445 (move decimal left 3 times).

Did we increase or decrease -9445.3? (absolute value)

We decreased the absolute value by 3 decimal places, so we must increase the characteristic

955 x 108 becomes 9.55 x 1010

Your Turn

1. 36.7 x 101

2. -0.015 x 103

3. 75.4 x 10-1

4. -14.5 x 102

5. 0.123 x 104

6. 97723 x 10-2

7. 851.6 x 10-3

8. 94.2 x 10-4

9. -0.012 x 103

10. 966 x 10-1

3.76 x 102

-1.5 x 10-5

7.54 x 100

May drop leading zeros - keep trailing

-1.45 x 103

1.23 x 103

9.7723 x 102

8.516 x 10-3

9.42 x 10-4

-1.2 x 101

9.66 x 10-1

Rule for MultiplicationCalculating with Numbers Written in Scientific Notation

1. MULTIPLY the mantissas2. Algebraically ADD the characteristics3. Correct the result to proper scientific notation when needed

Sample Problem: (4 x 10-3) (3 x 10-3)

(4) x (3) = 12

(-3) + (4) = 1 or 101

1.2 x 102

Rule for DivisionCalculating with Numbers Written in Scientific Notation

1. DIVIDE the mantissas2. SUBTRACT the characteristic of the denominator from

the characteristic of the numerator3. Correct the result to proper sci. notation if needed

Sample Problem: Divide 7.2 x 10-4 by -8 x 105

(7.2) (-8) = -0.9

(-4) - 5) = -1 or 10-9

-0.9 x 10-9

Correct: -9 x 10-10

.

.

Your Turn

1. (2 x 104)(3 x 10-3)2. (5 x 10-3)(4 x 10-4)3. (6 x104)(-7 x 10-5)4. (-4.5 x 10-2)(2 x 10-7)

5. (8 x 10-5) / (2 x 10-3)6. (4 x 103) / (8 x 10-3)7. (6 x 10-7)/(3 x 108)8. (4.5 x 104) / (9.0 X 10-12)9. ((2 X 103)(4X10-2)) / ((6 X 10-9)(4 X 105))

6 x 101

20 X 10-7 2 x 10-6

-42 x 10-1 4.2 x 100

May drop leading zeros - keep trailing

-0 x 10-9

4 x 10-2

5.0 x 105

2 x 101

5 x 1015

3.3 x 104

Rule for Addition and SubtractionCalculating with Numbers Written in Scientific Notation

In order to add or subtract numbers written in scientific notation, you must express them with the same power of 10.(Same characteristic). Then correct to proper scientific notation.

Sample Problem: Add 5.8 x 103 and 2.16 x 104

(5.8 x 103) + (21.6 x 103) = 27.4 x 103

Exercise: Add 8.32 x 10-7 and 1.2 x 10-5 1.28 x 10-5

2.74 x 104