S. H. Unger 8/12/99

Technical professionals should be comfortable manipulating numbers and presenting numerical results clearly. They should be adept with calculators, but should also be able to do calculations manually and mentally. Following are a few important points that should be well understood.

1. Scientific notation.

The generally accepted way to present numbers, particularly large or small numbers, is to write them as a decimal number d, where 1<= d < 10, multiplied by a power of 10. (In the following , exponentiation is indicated by ^, e.g., 2^3 represents 2 cubed.) For example, 3.725 x 10^17 or 5.6 x 10^-12. In many situations, where the number is not extremely large or extremely small, we write it out in the conventional manner. That is, we might write 356 rather than 3.56 x 10^2.

2. Accuracy and Precision.

In the real world, physical quantities are almost always approximations. We can't determine the EXACT value of a mass or speed or voltage or current. (Often--though not always--we CAN determine INTEGRAL values exactly. For example, we can count the number of memory chips on a circuit board, but it is usually not feasible to determine exactly the number of people in a large crowd.)

When we state a physical quantity, it is therefore necessary to indicate how close we claim it is to the "correct" value. We might, for example, state that we estimate a certain time interval to be in the range between 4.5 and 5.5 seconds. If it turns out that the actual value is within the given range, then our estimate was ACCURATE. So in the previous example, if the actual value is 4.7, then our estimate was accurate, while if the actual value was 5.8, our estimate was not accurate--it was incorrect. An alternative way of stating the same estimate is to say that we estimate the interval to be 5 seconds, with a tolerance of 10%, i.e., that our estimate may be off in either direction by up to 10 % (which is, of course, 0.5). If the tolerance is very small, then our estimate is very PRECISE. That is, a precise estimate is one with a small tolerance.

A common practice is to IMPLY precision via the number of digits in the cofactor of the power of ten in a scientific form number. So we might, for example, predict that the running time of a program will be 3 x 10^5 seconds. The implication here is that we claim the correct answer will be between 2.5 x 10^5 and 3.5 x 10^5 seconds. We are saying that our prediction is accurate to one significant figure, the 3 in 3 x 10^5. If we wanted to claim higher precision, we could have written our estimate as 3.00 x 10^5 seconds. Now we are asserting that our estimate is accurate to 3 significant figures, i.e., that the 3 digits in 3.00 are all correct, meaning that the actual running time will be between 2.995 and 3.005 x 10^-5 seconds.

It is poor practice to state results with precision that is clearly not justified. If, for example, physical data is obtained using instruments accurate to within 10%, then numbers resulting from computations based on such data should reflect the precision of the input numbers. For example, it would be appropriate to state a result based on such data as 6 x 10^5, but certainly not as 6.2345 x 10^5. The latter implies precision not justified by the data. (Sometimes one extra figure might be given to take into account the fact that the accuracy of the last digit varies depending on the magnitude of the most significant digit. For example, the accuracy of the third digit in 1.23 is almost the same as that of the second digit in 9.5. An error of one in each case corresponds to roughly 1 %.) A common practice among novices is to present blindly the full output of a calculator, when the input data was very crude, perhaps valid to only one significant figure. For example, if one divides two numbers, one of which is accurate to no more than 5 %, it is absurd to present the result as, 3.6579902. Calculator results should be rounded off to the appropriate number of digits. For example, if our results are considered to have a precision of 3 significant figures, we would round off a calculator result of 56789012, writing it as 5.68 x 10^7. (There is some slack here; 5.679 x 10^7 would not be unreasonable).

3. Standard Prefixes

It is always acceptable to state results as a number in scientific form modifying a basic physical unit. For example, one might write 3.06 x 10^11 watts. But we often use standard prefixes to replace the powers of ten. For example, the preceding quantity might also be specified as 306 gigawatts. Or 7.1 x 10^-8 second is often stated as 71 nanoseconds. The standard prefixes (with the associated powers of 10) are: kilo (3), mega (6), giga (9), tera (12), peta (15), exa (18), centi (-2), milli (-3), micro (-6), nano (-9), pico (-12), femto (-15), atto (-18). We seldom see the extreme units such as peta, exa, femto, and atto, but, as technology progresses, the likelihood is that they will be used more often. For example, as integrated circuit speeds increase, rather than talk about hundredths of a nanosecond, we now see terms such as 34 picoseconds. Rather than see .04 picofarads, we will see 40 femtofarads. A common practice is to use a prefix that allows the integer part of the associated scientific form number to have no more than 3 digits. So we would replace 1.23 x 10^5 watts with 123 KW (kilowatts), and 3.2 x 10^-3 amperes would be converted to 3.2 mA (milliamperes). (We often abbreviate the modified units as in the previous examples.)

4. Rational Fractions

In special cases where it is appropriate to state results as rational fractions, if the numerator and denominator are relatively small, the fractions should be reduced. That is, a number such as 27/12 should be presented as 9/4. It is sometimes acceptable or desirable to write such a number as 2 1/4.

5. Estimating Powers of 2.

In the computer field we often encounter situations where it is important to find an integer power of two. For example, the number of memory locations that can be accessed using a 14-bit address is 2^14. Often, the answer is a very large number. It is usually not necessary to find exact values--an answer off by less than 10 % is generally quite satisfactory. Computer professionals should be able to make such calculations quickly in their heads. Here is how to do it. First memorize the values of 2^n for n ranging from 1 to 10. These are, of course, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Note that 2^10 = 1024 is a close approximation to 10^3. It follows that 2^20 is about 10^6, 2^30 is about 10^9, etc. Now, given an arbitrary positive integer, such as 24, we can approximate 2^24 quickly by thinking of it as 2^4 x 2^20. Knowing that 2^4 = 16 and that 2^20 = 10^6, we can see that the answer is close to 16 million (1.6 x 10^7). Of course, this approximation is always less than the correct answer, but, the error remains under 10% for powers of 2 up to 40. Check this out! Note that, in recognition of this approximation, in the computer field, K is often used to represent 2^10, and M is used to represent 2^20. So, when a memory chip is said to store 64 K bits, we may approximate this as 64,000, but the actual number is 64 x 1024 (i.e., 65, 536). Note incidentally, that a common practice is to abbreviate bits with b and bytes by B (of course there are 8 bits in a byte). So we may see 16 MB for 16 million bytes or 40 Mb for 40 million bits.


Number Handling Problems

Test yourself on your ability to deal with numbers properly by solving the problems below. You must turn in your answers with the first HW assigment, altho they will not be graded. Solutions will be posted on the web site next week so you can check out how well you have done.

1. Express each of the following numbers in scientific form:

(a) 1234567

(b) 0.000000567

(c) 345.678

2. Express each of the following scientific form numbers as an ordinary decimal number.

(a) 3.665 x 10^4

(b) 1.009 x 10^-6

3. Round off each of the numbers in question 2 to 2 significant figures:

4. If, in a particular manufacturing process, resistors can be made with a tolerance of 10%, specify in scientific form, the value of a resistor that our computer program indicates should be 3456 ohms?

5. Specify 351/12 as a:

(a) reduced rational fraction

(b) sum of an integer and a reduced proper fraction

(c) decimal number with 3 significant figures

6. Approximate 2^n in scientific form for each of the following values of n:

(a) 13

(b) 29

(c) 27

(d) 34

7. (a) How many BITS are there (approximately) in a 32 MB memory?

(b) How many BYTES are there (approximately) in a 16 Kb memory?

8. (a) How many joules are there in 37 terajoules?

(b) How many femtofarads are equivalent to 0.034 picofarads?

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