...
Sometimes exact values are cumbersome and irrelevant.
Example: World population
Is that indeed accurate? --NO
Does one want an accurate number? --NO
One would better say: around 7 billion, or write 7.4×109
Q×a power of 10,
Q --- a quantity between 1 and 10 ,
a power of 10 is 10a whole number
Some powers of ten have their own names:
| 102 | | | hundred |
| 103 | | | thousand |
| 106 | | | million |
| 109 | | | billion |
| 1012 | | | trillion |
Instead of: "Total spending in the new federal budget is $3,900,000,000,000"
we have: "Total spending in the new federal budget is $3.9×1012", or
"Total spending in the new federal budget is $3.9 trillion".
Instead of: "The diameter of a typical bacteria is 0.000001 meter"
we have: "The diameter of a typical bacteria is 10−6 meter", or "The diameter of a typical bacteria is 1 micrometer".
| 10−1 | | | deci |
| 10−2 | | | centi |
| 10−3 | | | milli |
| 10−6 | | | micro |
| 10−9 | | | nano |
| 10−12 | | | pico |
Converting to scientific notation
Step 1 Move decimal point to come after the first digit
Step 2 Number of places moved is the power of ten
Examples:
3042 ...... 3 places to the left ...... 3.042×103Converting from scientific notation
Just move the decimal point in the opposite way
Examples:
3.042×103 ...... 3 places to the right ...... 3042Multiplication and division in scientific notation
Operate separately with the powers of 10 and the rest.
Examples:
(6×102)×(4×105) =(6×4)×(102×105) =24×107 =2.4×108Addition and subtraction in scientific notation
Either translate everything into ordinary notations, perform the operations, and translate back, or ... think a bit ...
Example: 7.4×109+9.5×103 ≈7.4×109
The number in this example has to do with the global population.
One does not know it to this accuracy, and does not really need this precision.
Putting numbers in perspective is our next topic to discuss
The book lists three methods:
Perspective through estimation
Perspective through comparison
Perspective through scaling
The idea is always to grasp the
order of magnitude,
which is the broad range, within one or two powers of ten,
with as small amount of calculations as possible.
Example (30, p.148) Could a person walk across the United States (New York to California) in a year?
Collecting data:
2,765 miles within 365 days ...
means less than 8 miles a day.
Answer. Yes, that is possible.
Meaning: very rough estimate, number of digits ...
Example (36, p.148)
The total number of words in the textbook.
About 700 pages, 50 rows on a page, 15 words in a row...
700=7×102 ... 35×103 ... 500×103=5×105
Reasonable answer: several hundred thousand.
Table 3.1 on p.140 tells us:
Energy released by burning 1 kilogram of coal is 1.6×109 joules.
Energy released by fission of 1 kilogram of Uranium-235 is 5.6×1013 joules.
Clearly, fission is more efficient, but how much?
1 kilogram of Uranium-235 gives same amount of energy as
5.6×10131.6×109=3.5×104, which is 35 tones (thousand kilos) of coal
Here I refer to Example 7 on p.142
The distance from the Earth to the Sun is 150 million kilometers, the diameter of the Sun is 1.5 million kilometers, while the diamemter of the Earth is 12,760 kilometers. ... So what?
After dividing all these lengths by 1010, they find out the following.
If the Sun was of the size of grapefruit, the Earth would be of size of a ball point in a pen at a distance of 15 meters from the grapefruit.