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: 6.022X10

23 602200000000000000000000

Method to express really big or small numbers.

Format is Mantissa x Base Power

Decimal part of original number Decimal you moved

6.02 x 10

23 We just move the decimal point around.

602000000000000000000000

Using the Exponent Key on a Calculator

EE EXP

EE or EXP means “times 10 to the…”

How to type out 6.02 x 10 23 :

: .

.

Don’t do it like this… 6

.

0 …or like this… 6

.

0 …or like this: 6

.

0 2 2

x x

2 1 y x 0 2 EE 3 WRONG!

WRONG!

2 3 1 TOO MUCH WORK.

0 y x 2 3

Example:

1.2 x 10 5 Type this calculation in like this:  2.8 x 10 13 1

.

2 EE 5  2

.

8 EE 1 3 Calculator gives… or…

4.2857143 –09 4.2857143 E –09

= This is NOT written… 4.3

–9 But instead is written… 4.3 x 10 –9 or 4.3 E –9

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 10 2  

Convert -0.00634

-6.34 x 10 -3

Your Turn

1. 17600.0

2. 0.00135

3. 10.2

4.

-67.30

5.

6.

4.76

-

0.1544

7.

8.

301.0

-0.000130

1.76 x 10 4 1.35 x 10 -3 1.02 x 10 1 -6.730 x 10 1 4.76 x 10 0 -1.544 x 10 -1 3.010 x 10 2 -1.30 x 10 -4 May drop leading zeros - keep trailing

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 10 5  -977,000

7.5

x 10 6  8.7

x 10 4 = -6.525 x 10 -9 report -6.5 x 10 -9 (2 sig. figs.) 4.35

x 10 6  1.23

x 10 3 = 5.3505 x 10 3 or 5350.5

report 5.35 x 10 3 (3 sig. figs.) 5.76

x 10 16  9.86

x 10 4 = 5.84178499 x 10 -13 report 5.84 x 10 -13 (3 sig. figs.) 8.8

x 10 11  3.3

x 10 11 6.022

x 10 23  5.1

x 10 8 = 2.904 x 10 23 report 2.9 x 10 23 (2 sig. figs.) = -3.07122 x 10 16 report -3.1 x 10 16 (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 10 8  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 10 8 becomes 9.55 x 10 10 -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 10 8 becomes 9.55 x 10 10

Your Turn

1. 36.7 x 10 1 2. -0.015 x 10 3 3. 75.4 x 10 -1 4. -14.5 x 10 2 5. 0.123 x 10 4 6. 97723 x 10 -2 7. 851.6 x 10 -3 8. 94.2 x 10 -4 9. -0.012 x 10 3 10. 966 x 10 -1

3.76 x 10 2 -1.5 x 10 -5 7.54 x 10 0 -1.45 x 10 3 1.23 x 10 3 9.7723 x 10 2 8.516 x 10 -3 9.42 x 10 -4 -1.2 x 10 1 9.66 x 10 -1 May drop leading zeros - keep trailing

Rule for Multiplication

Calculating with Numbers Written in Scientific Notation 1. MULTIPLY the mantissas 2. Algebraically ADD the characteristics 3. 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 10 1

1.2 x 10 2

Rule for Division

Calculating with Numbers Written in Scientific Notation 1. DIVIDE the mantissas 2. SUBTRACT the characteristic of the denominator from the characteristic of the numerator 3. Correct the result to proper sci. notation if needed Sample Problem: Divide 7.2 x 10 -4 by -8 x 10 5 ( 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 10

4

)(3 x 10

-3

) 2. (5 x 10

-3

)(4 x 10

-4

) 3. (6 x10

4

)(-7 x 10

-5

) 4. (-4.5 x 10 -2 )(2 x 10 -7 )

6 x 10 1 20 X 10 -7 -42 x 10 -1   2 x 10 -6 4.2 x 10 0 -0 x 10 -9 1.

2.

3.

4.

5.

(8 x 10 -5 ) / (2 x 10 -3 ) (4 x 10 3 ) / (8 x 10 -3 ) (6 x 10 -7 )/(3 x 10 8 ) (4.5 x 10 4 ) / (9.0 X 10 -12 ) ((2 X 10 3 )(4X10 -2 )) / ((6 X 10 -9 )(4 X 10 5 )) 4 x 10 -2 5.0 x 10 5 2 x 10 1 5 x 10 15 3.3 x 10 4 May drop leading zeros - keep trailing

Rule for Addition and Subtraction

Calculating 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 10 3 and 2.16

x 10 4 ( 5.8 x 10 3 ) + ( 21.6 x 10 3 ) = 27.4 x 10 3

2.74 x 10 4

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

1.28 x 10 -5