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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