General Astronomy/Scientific Notation
|Short History of the Universe||Scientific Notation||The Scientific Method|
In previous sections, we discussed some numbers that were very large. In astronomy, the appearance of such huge numbers is common. This is one reason astronomers and other scientists use scientific notation when working with very large or very small numbers. Scientific notation is a system for writing and working with numbers that makes it much easier to deal with numbers that are very small or very large.
For example, the Milky Way Galaxy contains roughly three duodecillion tons of material. That is a rather cumbersome number. (Astronomers would never actually write this. Instead, they would say that the Milky Way contains one trillion times the mass of the Sun, which is somewhat easier. We'll use this much larger number for our demonstration.) You could also write this number as
- 3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,
but that's even worse. Scientific notation makes the number much more compact and readable:
- 3 × 1039 tons.
This number is verbally expressed as "three times ten to the 39 power tons." This is numerically equivalent to the first two expressions.
A number written correctly in scientific notation has two parts. The first is a number greater than or equal to 1 and less than 10 (but it can be either positive or negative). This is sometimes called the mantissa. The second part is the number ten raised to a whole number power. The exponent of the second number is called the power. Some examples of numbers written correctly in scientific notation are:
- 2 × 1018
- -1.4 × 102
- 7.656 × 10-4
- 2.1 × 100
These, on the other hand, are not valid examples of numbers written in scientific notation:
- 0.1 × 104 is wrong because the mantissa is less than 1
- 12 × 103 is wrong because the mantissa is not less than 10
- 8.4 × 102.2 is wrong because the power is not a whole number
- 10n = 10 × 10 × 10 × ... for n times,
which means that ten raised to n power is the same as 10 multiplied by itself n times, which is the same as a 1 with n zeros written after it. For example, 103 is 10 × 10 × 10, or 1000. That means that our earlier number, 3 × 1039 tons, is equivalent to
- 3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,
which is a three followed by 39 zeros. A number written in scientific notation with a negative power corresponds to a small number. For example, the number 1 × 10−3 is written as 0.001 in conventional notation. In general,
- 10-n = 1/10 × 1/10 × 1/10 × ... for n times.
Since scientific notation relies on powers of ten, it's simple to convert a number from scientific notation to standard notation or vice versa. To convert a large number (with a positive power) from scientific notation to standard notation, first identify the decimal point in the mantissa, then shift the decimal to the right by the number indicated by the power. To convert a number from standard notation to scientific notation, just reverse these steps. Find the decimal point in the number, and move it until the number is at least 1 but less than 10. Count the number of places you moved the decimal point and use that number as the power. If you moved the decimal point to the left, make the power positive. If you moved the decimal point to the right, make the power negative.
Scientific notation also makes it simpler to do multiplication and division. To multiply two numbers in scientific notation, multiply the mantissas and add the powers:
- (3 × 104) × (4 × 10-2)
- (3 × 4) × 104 - 2
- 12 × 102
- 1.2 × 103
In some cases, such as the one shown here, you may need to shift the decimal point again ensure that the number is in correct scientific notation. It should never be necessary to shift the decimal point by more than one digit. When dividing numbers in scientific notation, divide the mantissas and subtract the powers:
- 0.75 × 106
- 7.5 × 105
Here also, it may be necessary to shift the decimal point and change the exponent.
Scientific notation makes it easy to compare numbers that have very different values because all the zeroes have been replaced with the much more readable exponent. Numbers with a greater exponent are always bigger than numbers with a lesser exponent.
If one of the exponents is bigger than the other by more than a couple, the difference between the two is clearly very big. Recognizing a huge difference between two numbers can sometimes be a very useful insight, so it often makes sense to take a moment to develop an intuitive feel for a math problem before attacking it. In some cases, it's useful to see roughly by how much one number is larger than another. Scientific notation makes this much simpler. For a rough estimate, you only need to find the difference in the exponents. For example, 107 is greater than 103, since 7 - 3 = 4.
- Some tourists in the Chicago Museum of Natural History are marveling at the dinosaur bones. One of them asks the guard, "Can you tell me how old the dinosaur bones are?"
- The guard replies, "They are 73 million, four years, and six months old."
- "That's an awfully exact number," says the tourist. "How do you know their age so precisely?"
- The guard answers, "Well, the dinosaur bones were seventy three million years old when I started working here, and that was four and a half years ago."
- (From the Science Jokes Web page )
In science, measurements are never perfect and numbers are never exact. As a result, every measurement we make has some uncertainty associated with it. Scientific notation makes it easy to express how precisely a number is known. Suppose a paleontologist discovers ancient dinosaur bones and finds that they are 73 million years old. Of course, the paleontologist doesn't know exactly how old they are. Maybe they're 73,124,987 years old, but the paleontologist only knows the age within 1 million years, so the age is written as 73,000,000 years, or 7.3 × 107 years. Either of these expressions imply that the bones aren't exactly 73 million years old, but are 73 million years old, give or take a million years.
But what if the paleontologist knows the age within 200,000 years, and is sure that the bones aren't, say, 73.4 million years old? In that case, the standard notation is ambiguous — the number is still written as 73,000,000 years. In scientific notation, we can write the number as 7.30 × 107 years. If we write this, we mean that the third digit is significant. The paleontologist might have calculated that the bones are 72,954,332 years old, but it would be useless to report these numbers, since the error on this measurement was 200,000 years. The extra digits are insignificant. The number of significant figures in a number are a reflection of the precision expressed in the number. In this case, the number of significant figures is three. The first significant figure is 7, the second is 3, and the third is 0.
Scientific notation gives the trailing digits written after the decimal point of a number special meaning — they tell that the number is exactly 7.30 × 107 years. This is unlike the usual use of numbers in mathematics, where trailing zeroes after the decimal point have no special meaning.
In the story about the museum guard, the guard hadn't thought about the precision of the age of the bones. It doesn't make sense to add the four years to 73 million years because the uncertainty in the age quoted to the guard was a lot more than 4 years. When working with numbers that have uncertainties, we must be sure not to express better precision in the results of our arithmetic than we had in the first place.
When doing arithmetic, numbers being added or subtracted are treated differently from numbers being multiplied or divided.
- When multiplying or dividing numbers with uncertainties, make sure that the answer has as many significant figures as the least precise of the original numbers.
For example, in (2.3 × 103) × (1.21 × 102), the number 2.3 × 103 has two significant figures and the number 1.21 × 102 has three significant figures. The result should have two significant figures: 2.8 × 105. We assume that there was some uncertainty in the measurement that gave us 2.3 × 103, and this leads to some uncertainty in the result of the computation.
Addition and subtraction work differently. When adding 23.14 and 2.2, for example, the number 2.2 has uncertainty beginning in the tenths place. This uncertainty makes it pointless to report the hundredths place in the sum. To see this, try adding some uncertainty to 2.2 and see how this affects the sum.
- When adding or subtracting numbers with uncertainties, round the result to the last significant place of the original number with the greatest uncertainty.
For example, 2.3 × 103 + 1.1 × 102 can be written as
2300 + 110 2410
But we don't know the real value of the tens place in 2.3 × 103, so we really only know the answer to the hundreds place. We should write
2300 + 110 2400
or 2.3 × 103 + 1.1 × 102 = 2.4 × 103. This may seem incorrect, but we're really only rounding off. Since we don't know the result to better than two significant figures, it makes no sense to report the extra digits — that would be like the museum guard who tells visitors that the dinosaur bones are 73 million and four years, six months old.
Almost every number has a unit of measurement attached to it. The number we used as our first example carried units of tons, and we expressed the age of the dinosaur bones in years. The units a number carries are part of the number itself. Units can also be multiplied or divided just like numbers.
As an example, consider a simple equation:
- distance = velocity × time.
Suppose you drive in a car with a speed of 100 kilometers per hour (60 miles per hour) and you go in a straight line for one hour. The distance you will travel is
or 60 miles. We have cancelled hours as though it were a number.
This trick is also useful when you need to convert units. If you have a result in one system and you'd like to convert to another, you can set up a ratio, such as 1,000 meters/1 kilometer. Since 1,000 meters is equal to 1 kilometer, the ratio 1,000 meters/1 kilometer equals one. Because of this, multiplying any number by 1,000 meters / 1 kilometer will not change the value of the number. If we want to know what 100 kilometers is in meters, we can write
or 100,000 meters.
Other units of measurement used in astronomy are kilograms (mass), Newtons (force), and Joules (energy).
|Short History of the Universe||Scientifc Notation||The Scientific Method|