Scientific Notation- the VERY BIG and the very small

Converting to Scientific Notation

Scientific notation is a way of easily and conveniently dealing with very large or very small numbers.

Consider the mass of an electron:

0.000 000 000 000 000 000 000 000 000 000 910 kilograms

and the number of seconds in 1 year:

3,153,600,000 seconds

The Very Large

Let's check out this number:

3,153,600,000 seconds

3.1536 x 10

31.536 x 10315.36 x 10

3153.6 x 10

31536 x 10

315360 x 10

3153600 x 10

31536000 x 10

315360000 x 10

31536000003.1536 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 =

= 3.1536 x 109

The Very Small

0.000 000 000 000 000 000 000 000 000 000 910 kilograms

Converting from Standard form to Scientific Notation

1) Find the decimal place.

2) Start at the left side of the number and put a box around the first number you come to. (This number will be between 1 and 9)

3) Put a new decimal point to the right of the box.

4) Rewrite this new number

5) Put a "x 10" after the new number

4) Count the number of places between the old decimal place and the new one.

5) Put this number as the exponent. (Remember + if you have to go to the left and - if you have to go right.)

0.000 000 000 000 000 000 000 000 000 000 910 kilograms

3,153,600,000 seconds

What if the number already has an exponent?

347.66 x 104

Try these:

a) 7, 960,000,000

b) 0.007485

c) 45.756

d) 998.6 x 102

e) 64.3 x 10-4

Convert to standard form

3.65 x 104 seconds =

8.877 x 10 -3 metres =

The numerical values we deal with in science represent measurements whose values are never known exactly.

CALCULATORS ARE STUPID

-Calculators treat the numbers we punch into them as "pure" mathematical entities.

-The result is that they frequently give answers that are physically ridiculous even though mathematically correct.

CALCULATORS ARE STUPID!!

They do not ask if an answer makes sense.

That is your job!

Significant Digits

Important point:

-the concept of significant digits has less to do with mathematics than with our confidence in a measurement.

-When you measure something you are comparing it to something else.

-How many significant digits we have will depend on what we are measuring and what instrument we are using to do the measuring.

• Number  of  significant  figures  =  number  of  digits  we  are  certain  of  plus  one  final  uncertain  digit.• Each  measurement  keeps  only  one  uncertain  digit.

Significant Digits or Significant Figures

Significant Digits Rules

• All  non-­‐zero  digits  are  significant.• All  zeros  between  non-­‐zero  digits  are  significant.• Leading  zeros  in  decimals  are  not  significant.• Trailing  zeros  are  significant  if  they  are  to  the  right  of  a  decimal.• In  numbers  greater  than  1,  trailing  zeros  are  not  significant  unless  they  are  stated  to  be  so.

Memorize These Rules

• Try  this  addition.    Do  not  use  a  calculator.5.4  m    3.45  m    +0.656  m

??

Adding and Subtracting

What  do  you  think  the  rule  for  Addition  and  Subtraction  might  be?

In  each  case  the  least  significant  digit  is  in  red.

• The  sum  or  difference  of  an  addition  or  subtraction  equation  is  only  as  accurate  as  the  least  accurate  measurement.  

• In  other  words  if  a  final  uncertain  digit  is  combined  with  a  certain  digit  the  result  becomes  uncertain  itself.

• The  answer  to  a  calculation  keeps  only  one  uncertain  digit.

Rules For Adding and Subtraction

Try these:

1) 1.34 cm + 2.5 cm =

2) 34.557 m/s - 0.08 m/s =

3) 14.46 kg - 8 kg =

• Try  this  multiplication.    Do  not  use  a  calculator.14.25x6.43

Multiplication and Division

Can  you  think  of  the  rule  for  multiplication?

The  least  significant  digit  is  in  red.

• The  resultant  (i.e.  product  in  our  example)  is  only  as  accurate  as  the  factor  with  the  least  number  of  significant  digits.

• In  other  words  if  the  final  uncertain  digit  is  multiplied  or  divided  by  a  certain  digit  the  result  becomes  uncertain.

• Answers  to  calculations  keep  only  one  uncertain  digit.

Rules For Multiplication and Division

Try these:

1) 7.64 x .5 =

2) 154.778 ÷ .33 =

3) 6.54 x 102 x 3.00 =

• If  the  number  to  be  rounded  off  is  greater  than  of  equal  to  5,  the  next  digit  is  increased  by  1.

• If  the  digit  to  be  dropped  is  less  than  5,  the  preceding  digit  remains  the  same.

• If  the  digits  to  be  dropped  are  to  the  left  of  the  decimal  point  and  zeros  are  used  as  their  replacements  express  your  answer  in  scientific  notation.

As  a  general  rule  only  round  off  at  the  END  of  a  series  of  calculations.

Rules For Rounding Off

Distance between the Sun and Earth = 149,000,000,000 meters

Size of a Hydrogen Atom = 0.000000000053 meters

http://www.khanacademy.org/video/scientific-notation-1?playlist=Algebra%20I%20Worked%20Examples

http://www.khanacademy.org/video/scientific-notation-2?playlist=Algebra%20I%20Worked%20Examples

http://www.khanacademy.org/video/scientific-notation-3?playlist=Algebra%20I%20Worked%20Examples

Or just go to khanacademy.org and search through for scientific notation.

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