| 25 | Scientific Notation Tutorial |
In science and engineering, huge or tiny quantities can be unwieldy when written out in full as decimal numerals. Scientific notation provides a “shortcut” for writing such quantities. You’ve already seen scientific notation in this book. Chapter 24 had plenty of it! If any of this notation has confused you in the past, this chapter explains and reviews how scientific notation and significant figures work.
STANDARD FORM
The scientist’s and engineer’s way of denoting extreme quantities is to write them as real number multiples of integer powers of 10. A numeral in standard scientific notation, also called the American form, is written as follows:
where the dot (.) is a period written on the base-line, representing the decimal point. The numeral m to the left of the radix point is a single-digit positive or negative integer (it can never be 0). Each of the numerals n1, n2, n3, and so on up to np to the right of the radix point is a single-digit nonnegative integer (it can be 0). The entire decimal expression to the left of the multiplication symbol is called the coefficient. The value z, which is the power of 10, can be any integer. Here are three examples of numbers written in standard scientific notation:
When writing numbers in scientific notation, you must take care to be sure the form is right. For example, take a close look at 245.89 × 108. This represents a perfectly legitimate number, but it is not written in the correct form for scientific notation. The number to the left of the multiplication symbol must always be a single digit from the set {−9, −8,−7, −6 −5, −4, −3, −2, −1, 1, 2, 3, 4, 5, 6, 7, 8, 9}. To write down this particular number in the proper format for scientific notation, first divide the portion to the left of the multiplication symbol by 100, so it becomes 2.4589. Then multiply the portion to the right of the multiplication symbol by 100, increasing the exponent by 2 so it becomes 1010. This produces the same numerical value but in the correct format for standard scientific notation: 2.4589 × 1010.
ALTERNATIVE FORM
In some literature, a variation on the above theme is used. You can call it alternative scientific notation. Some people call it the European form. This system requires that the number m to the left of the radix point always be equal to 0. When numbers are expressed this way, the exponent is increased by 1 compared with the same number in standard scientific notation. In alternative scientific notation, the above three quantities would be expressed like this:
Note that when a negative exponent is “increased by 1,” it becomes “1 less negatively.” For example, when −15 is increased by 1, it becomes −14, not −16.
THE “TIMES” SIGN
The multiplication sign in scientific notation can be denoted in various ways. Most scientists in the United States use the cross symbol (×), as in the examples shown above. But a small “center” dot raised above the base-line (.) can also be used to represent multiplication in scientific notation. When written that way, the above numbers look like this in the standard power-of-10 form:
This small dot should not be confused with a radix point.
A small dot symbol is preferred when multiplication is required to express the dimensions of a physical unit. An example is the kilogram-meter per second squared, which is symbolized kg · m/s2 or kg · m · s−2. As you’ve already seen, the small dot is also commonly used when two simple numerals are multiplied. For example,
Another alternative multiplication symbol in scientific notation is the asterisk (*). You will occasionally see numbers written like this in standard scientific notation:
PLAIN-TEXT EXPONENTS
Once in awhile, you will have to express numbers in scientific notation using plain, unformatted text. This is the case when you are transmitting information within the body of an e-mail message (rather than as an attachment). Some calculators and computers use this system in their displays. An uppercase letter E indicates that the quantity immediately before it is to be multiplied by a power of 10, and that power is written immediately after the E. In this format, the above quantities are written
Still another alternative is the use of an asterisk to indicate multiplication, and the symbol ^ to indicate a superscript, so the expressions look like this:
In all these examples, the numerical values represented are identical. Respectively, if written out in full, they are
ORDERS OF MAGNITUDE
You learned a little about orders of magnitude in the last chapter. Let’s take a closer look at this concept. Consider the following two extreme numbers:
Now look at these:
Concerning the first number in each pair, the exponent 92 is 3 larger than 89. Considering the second number in each pair, the exponent −318 is 3 larger than −321. (Again, remember that numbers grow larger in the mathematical sense as they become more positive or less negative!) Both of the numbers in the second pair are therefore 3 orders of magnitude larger than their counterparts in the first pair. That’s a factor of 103, or 1,000.
The order-of-magnitude concept lets you create number lines and graphs that cover huge spans. Examples are shown in Fig. 25-1. Drawing A shows a log-scale number line that covers 3 orders of magnitude from 100 (which is equal to 1) to 103 (which is 1,000). Illustration B shows a log-scale number line spanning 10 orders of magnitude from 10−3 (0.001) to 107 (10,000,000). Illustration C shows a coordinate system with a log-scale horizontal axis spanning 10 orders of magnitude from 10−3 to 107, and a linear-scale vertical axis spanning values from 0 to 10.
Figure 25-1
At A, a number line spanning 3 orders of magnitude. At B, a number line spanning 10 orders of magnitude. At C, a coordinate system whose horizontal scale spans 10 orders of magnitude, and whose vertical scale is linear and extends from 0 to 10.
WHEN TO USE SCIENTIFIC NOTATION
In most technical documents, scientific notation is used only when the power of 10 is larger or smaller than certain values. If the power of 10 is between −2 and 2 inclusive, numbers are written out in plain decimal form as a rule. If the power of 10 is −3 or 3, numbers may appear in plain decimal form or in scientific notation. If the exponent is −4 or smaller, or if it is 4 or larger, values are expressed in scientific notation as a rule. In number lines and graphs, exceptions are sometimes made for consistency, as is done in Fig. 25-1.
Some calculators, when set for scientific notation, display all numbers that way, even when it is not necessary. This can be confusing, especially when the power of 10 is 0 (or close to it) and the calculator is set to display a lot of digits. Most people understand the expression 8.407 more easily than 8.407000000E+00, for example, even though both represent the same number.
PREFIX MULTIPLIERS
Special verbal prefixes and abbreviations, known as prefix multipliers, are used in the physical sciences and in engineering to express certain powers of 10. Table 25-1 shows these multipliers and abbreviations for factors ranging form 10−24 to 1024.
Table 25-1
Power-of-10 prefix multipliers and their symbols.
Let’s take an example. By how many orders of magnitude does a gigahertz differ from a kilohertz? (The hertz is a unit of frequency, equivalent to a cycle per second.) Using Table 25-1, you can figure out that a gigahertz (GHz) represents 109 hertz (Hz), and a kilohertz (kHz) represents 103 Hz. The exponents differ by 6. That means 1 GHz differs from 1 kHz by 6 orders of magnitude, which is a factor of 106, or 1,000,000.
Here’s another example. By how many orders of magnitude is a nanometer different from a centimeter? Look at the table again. The prefix multiplier nano- represents 10−9, and centi-represents 10−2. The exponents −9 and −2 differ by 7. The centimeter is the bigger unit of linear dimension. So a hair measuring a centimeter (1 cm) long is 7 orders of magnitude larger in linear dimension than a submicroscopic wire that is a nanometer (1 nm) long. That’s a factor of 107, or 10,000,000.
MULTIPLICATION
When you want to multiply two numbers in scientific notation, first multiply the coefficients by each other. Then add the exponents. Finally, reduce the expression to standard form if it doesn’t happen to be that way already. Here are three examples:
This last number can be written out in plain decimal form because the exponent is between −2 and 2 inclusive.
Here is the generalized rule for multiplication in scientific notation, using the variables u and v to represent the coefficients and the variables m and n to represent the exponents. Let u and v be real numbers greater than or equal to 1 but less than 10, and let m and n be integers. Then
DIVISION
When you want to divide numbers in scientific notation, first divide the coefficients by each other. Then subtract the exponent in the denominator from the exponent in the numerator. Finally, reduce the expression to standard scientific notation if it doesn’t happen to be that way already. Here are three examples of how division is done in scientific notation:
The numbers here do not divide out neatly, so the decimal-format portions are approximated.
Here’s the generalized rule for division in scientific notation, using the variables u and v to represent the coefficients and the variables m and n to represent the exponents. Let u and v be real numbers greater than or equal to 1 but less than 10, and let m and n be integers. Then
RAISING TO A POWER
When a number is raised to a power in scientific notation, both the coefficient and the power of 10 itself must be raised to that power, and the result multiplied. Here is an example:
Now look at another example, in which the power of 10 is negative:
Here’s the generalized rule for taking a number to a power in scientific notation, using the variable u to represent the coefficient and the variables m and n to represent the exponents. Let u be a real number greater than or equal to 1 but less than 10, and let m and n be integers. Then
TAKING A ROOT
To find the root of a number in scientific notation, think of it as a fractional exponent. The positive square root is equivalent to the 1/2 power. The cube root is the same thing as the 1/3 power. In general, the positive nth root of a number (where n is a positive integer) is the same thing as the 1/n power. When you think of roots in this way, you can multiply things out as you do with whole-number exponents. For example,
Here’s the generalized rule for taking the root of a number in scientific notation, using the variable u to represent the coefficient and the variables m and n to represent the exponents. Let u be a real number equal to at least 1 but less than 10. Let m and n be integers. Then
ADDITION
Scientific notation can be awkward when you are adding sums, unless all the addends are expressed to the same power of 10. Here are three examples:
SUBTRACTION
Subtraction follows the same basic rules as addition. It helps to convert the numbers to ordinary decimal format before subtracting.
SIGNIFICANT FIGURES
The number of significant figures (or digits) in an expression indicates the degree of accuracy to which you know a numerical value, or to which you have measured, or can measure, a quantity.
When you do multiplication, division, or exponentiation using scientific notation, the number of significant figures in the final calculation result cannot “legally” be greater than the number of significant figures in the least exact expression. Consider the two numbers x = 2.453 × 104 and y = 7.2 × 107. The following is a perfectly valid statement if the numerical values are exact:
But if x and y represent measured quantities, as is nearly always the case in experimental science and engineering, the above statement needs qualification. You must pay close attention to how much accuracy you claim.
HOW ACCURATE ARE YOU?
When you see a product or quotient containing quantities expressed in scientific notation, count the number of single digits in the coefficients of each number. Then take the smallest number of digits. This is the number of significant figures you can claim in the final answer or solution.
In the above example, there are four single digits in the coefficient of x and two single digits in the coefficient of y. So you must round off the answer, which appears to contain six significant figures, to two significant figures. It is important to use rounding, and not truncation, as follows:
In situations of this sort, if you insist on being “mathematically rigorous,” you can use approximate-equality symbols (the “squiggly” ones) throughout, because you are always dealing with approximate values. But most folks are content to use ordinary equality symbols. It is universally understood that physical measurements are inherently inexact. So from now on, let’s dispense with the “squigglies”!
Suppose you want to find the quotient x/y instead of the product xy. Proceed as follows:
Again, you must round off to two significant figures at the end. It’s best to keep a few extra digits during a calculation to minimize cumulative rounding errors. You saw the results of such errors in some of the problems in Chap. 24.
WHAT ABOUT 0?
Sometimes, when you make a calculation, you’ll get an answer that lands on a neat, seemingly whole-number value. Consider x = 1.41421 and y = 1.41422. Both of these have six significant figures. Taking significant figures into account, we have
This appears to be exactly equal to 2. But in the real world (the measurement of a physical quantity, for example), the presence of five zeros after the radix point indicates an uncertainty of up to plus or minus 0.000005 (written ±0.000005). When you claim a certain number of significant figures, 0 is as important as any other digit.
Now find the product of 1.001 × 105 and 9.9 × 10−6, taking significant figures into account. First, multiply the coefficients and the powers of 10 separately, and then proceed from there, like this:
You must round this off to two significant figures, because that is the most you can legitimately claim. This particular expression does not have to be written out in power-of-10 form, because the exponent is within the range ±2 inclusive. Therefore,
WHAT ABOUT EXACT VALUES?
Once in awhile, some of or all the values in physical formulas are intended to be exact. An example is the equation for the area of a triangle
where A is the area, b is the base length, and h is the height. In this formula, 2 is a mathematical constant, and its value is exact. It can therefore have as many significant figures as you want, depending on the number of significant figures you are given in the initial statement of the problem.
Sometimes you’ll come across constants whose values are exact in a theoretical sense, but which you must round off when you want to assign them a certain number of significant figures. Two common examples are the irrational numbers π and e. Both of these are nonterminating, nonrepeating decimals, and they can never be exactly written down in that form. You have to round them off!
Expressed to 10 significant figures and then progressively rounded off (not truncated!) to fewer and fewer significant figures, π has values as follows:
You can use however many significant figures you need when you encounter a constant of this type in a formula.
MORE ABOUT ADDITION AND SUBTRACTION
When you add or subtract physical quantities that you have measured (in a lab, for example), determining the number of significant figures can involve subjective judgment. You can minimize the confusion by expanding all the values out to their plain decimal form, making the calculations, and then, at the end of the process, deciding how many significant figures you can claim.
Sometimes, the outcome of determining significant figures in a sum or difference is similar to what happens with multiplication or division. Take this sum x + y:
This calculation proceeds as follows:
Occasionally, one of the values in a sum or difference is insignificant with respect to the other. Suppose you want to find this sum x + y:
You calculate this way:
Here, y is so much smaller than x that it does not significantly affect the value of the sum. When you round off, you can only go to five significant figures, so the second value vanishes! You can say that the sum is the same as the larger number
If this confuses you, imagine that you set a U.S. dime on the back of an elephant. Does the elephant weigh significantly more with the dime than without it? No! But if you set a U.S. dime on top of a U.S. nickel, the pair weighs significantly more than the nickel alone.
QUESTIONS AND PROBLEMS
This is an open-book quiz. You may refer to the text in this chapter (and earlier ones, too, if you want) when figuring out the answers. Take your time! The correct answers are in the back of the book.
1. Write down the number 238,200,000,000,000 in scientific notation.
2. Write down the number 0.00000000678 in scientific notation.
3. Draw a number line that spans 4 orders of magnitude, from 1 to 104.
4. Draw a coordinate system with a horizontal scale that spans 2 orders of magnitude, from 1 to 100, and a vertical scale that spans 4 orders of magnitude, from 0.01 to 100.
5. What does 3.5562E+99 represent? How does it differ from 3.5562E−99?
6. Find the product of 8.0402 × 1064 and 2.73 × 10−63. Round the answer off to the largest justifiable number of significant figures, and express it properly.
7. Find the quotient −6.7888 × 1034 divided by 8.45453 × 1036. Round the answer off to the largest justifiable number of significant figures, and express it properly.
8. Calculate the 5th power of 4.57 × 107. Round the answer off to the largest justifiable number of significant figures, and express it properly.
9. Calculate the 4th root of 8.84 × 108. Round the answer off to the largest justifiable number of significant figures, and express it properly.
10. Find (5.33 × 1034) − (1.99 × 10−83). Round the answer off to the largest justifiable number of significant figures, and express it properly.
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