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![[Post New]](/templates/default/images/icon_minipost_new.gif) 23 Jan 2007 15:04:29 IST
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SIR I WANT 2 KNOW THE MEANING OF NEAREST TENTH? WANT 2 KNOW ABOUT RULES OF SIGNIFICANT FIGURES AND ROUNDING OF. ALSO RULES OF SCIENTIFIC NOTATIONS.
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AJEET |
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23 Jan 2007 20:54:03 IST
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SCIENTIFIC NOTATIONS:
Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10-9. So, how does this work? We can think of 5.6 x 10-9 as the product of two numbers: 5.6 (the digit term) and 10-9 (the exponential term). Here are some examples of scientific notation. | 10000 = 1 x 104 | 24327 = 2.4327 x 104 | | 1000 = 1 x 103 | 7354 = 7.354 x 103 | | 100 = 1 x 102 | 482 = 4.82 x 102 | | 10 = 1 x 101 | 89 = 8.9 x 101 (not usually done) | | 1 = 100 |
| | 1/10 = 0.1 = 1 x 10-1 | 0.32 = 3.2 x 10-1 (not usually done) | | 1/100 = 0.01 = 1 x 10-2 | 0.053 = 5.3 x 10-2 | | 1/1000 = 0.001 = 1 x 10-3 | 0.0078 = 7.8 x 10-3 | | 1/10000 = 0.0001 = 1 x 10-4 | 0.00044 = 4.4 x 10-4 | As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left. In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,
46600000 = 4.66 x 107 This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example, 0.00053 = 5.3 x 10-4 This number has 2 significant figures. The zeros are only place holders.
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The Scientist does not study nature because it is useful; he studies it because he delights in it, & he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, life would not be worth living. Ofcourse I do not here speak of that beauty that strikes the senses, the beauty of qualities & appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmoniuos order of the parts, & which a pure intelligence can grasp. |
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23 Jan 2007 20:56:19 IST
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Rules for Working with Significant Figures: - Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is specified.
Hint: Change the number to scientific notation. It is easier to see. - Addition or Subtraction:
The last digit retained is set by the first doubtful digit. - Multiplication or Division:
The answer contains no more significant figures than the least accurately known number.
EXAMPLES:
| Example | Number of
Significant Figures | Scientific Notation |
| 0.00682 | 3 | 6.82 x 10-3 | Leading zeros are not significant. |
| 1.072 | 4 | 1.072 (x 100) | Imbedded zeros are always significant. |
| 300 | 1 | 3 x 102 | Trailing zeros are significant only if the decimal point is specified. |
| 300. | 3 | 3.00 x 102 |
| 300.0 | 4 | 3.000 x 102 | EXAMPLES | Addition |  | Even though your calculator gives you the answer 8.0372, you must round off to 8.04. Your answer must only contain 1 doubtful number. Note that the doubtful digits are underlined. | | Subtraction |  | Subtraction is interesting when concerned with significant figures. Even though both numbers involved in the subtraction have 5 significant figures, the answer only has 3 significant figures when rounded correctly. Remember, the answer must only have 1 doubtful digit. | | Multiplication |  | The answer must be rounded off to 2 significant figures, since 1.6 only has 2 significant figures. | | Division |  | The answer must be rounded off to 3 significant figures, since 45.2 has only 3 significant figures. | Notes on Rounding - When rounding off numbers to a certain number of significant figures, do so to the nearest value.
- example: Round to 3 significant figures: 2.3467 x 104 (Answer: 2.35 x 104)
- example: Round to 2 significant figures: 1.612 x 103 (Answer: 1.6 x 103)
- What happens if there is a 5? There is an arbitrary rule:
- If the number before the 5 is odd, round up.
- If the number before the 5 is even, let it be.
The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out. - example: Round to 2 significant figures: 2.35 x 102 (Answer: 2.4 x 102)
- example: Round to 2 significant figures: 2.45 x 102 (Answer: 2.4 x 102)
- Of course, if we round to 2 significant figures: 2.451 x 102, the answer is definitely 2.5 x 102 since 2.451 x 102 is closer to 2.5 x 102 than 2.4 x 102.
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The Scientist does not study nature because it is useful; he studies it because he delights in it, & he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, life would not be worth living. Ofcourse I do not here speak of that beauty that strikes the senses, the beauty of qualities & appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmoniuos order of the parts, & which a pure intelligence can grasp. |
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23 Jan 2007 21:02:41 IST
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Rounding Decimals to the Nearest Tenth Rounding decimals is very similar to rounding other numbers. If the hundredths and thousandths places of a decimal is forty-nine or less, they are dropped and the tenths place does not change. For example, rounding 0.843 to the nearest tenth would give 0.8. If the hundredths and thousandths places are fifty or more, the tenths place is increased by one. The decimal 0.866 rounded to the nearest tenth is 0.9
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The Scientist does not study nature because it is useful; he studies it because he delights in it, & he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, life would not be worth living. Ofcourse I do not here speak of that beauty that strikes the senses, the beauty of qualities & appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmoniuos order of the parts, & which a pure intelligence can grasp. |
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