Bibliografía y referencias
Bloque 3: Comunicación escrita: escribir
When measured quantities are added or subtracted, determining the number of significant figures can involve subjective judgment. One way to resolve this is to expand all the values out to their plain decimal form (if possible), make the calculation, and then, at the end of the process, decide how many significant fig- ures you can reasonably claim.
In some cases, the outcome of determining significant figures in a sum or dif- ference is similar to what happens with multiplication or division. Take, for example, the sum x+ y, where x =3.778800 × 10−6 and y =9.22× 10−7. This
calculation proceeds as follows:
x=0.000003778800 y=0.000000922 x+y=0.0000047008
=4.7008×10−6 =4.70×10−6
In other instances, one of the values in a sum or difference is insignificant with respect to the other. Suppose that x=3.778800×104, while y=9.22×10−7.
The process of finding the sum goes like this: x=37,788 y=0.000000922 x+y=37,788.000000922
=3.7788000000922×104
In this case, y is so much smaller than x that it does not significantly affect the value of the sum. We can conclude that the sum here is the same as the larger number:
x+y=3.7788×104 PROBLEM 3-6
What is the product of 1.001 ×105 and 9.9 × 10−6, taking significant
SOLUTION 3-6
Multiply the coefficients and the powers of 10 separately: (1.001×105)(9.9×10−6)=(1.001×9.9)×(105×10−6)
=9.9099×10−1 =0.99099
We must round this to two significant figures, because that is the most we 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:
(1.001×105)(9.9×10−6)=0.99
Quick Practice
Here are some practice problems that cover the material presented in this chap- ter. Solutions follow problems.
PROBLEMS
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?
SOLUTIONS
1. This is the equivalent of 2.382 multiplied by 100,000,000,000,000 (or 1014), so in scientific notation, it is written as 2.382 ×1014.
2. This is the equivalent of 6.78 multiplied by 0.000000001 (or 10-9), so in
scientific notation, it is written as 6.78 ×10−9.
3. See Fig. 3-2.
1 10 102 103 104
Fig. 3-2. Illustration for Quick Practice Problem and Solution 3.
4. See Fig. 3-3. 1 10 100 0.01 0.1 1 10 100
Fig. 3-3. Illustration for Quick Practice Problem and Solution 4.
5. The first expression represents 3.5562 ×1099, which is a huge positive inte-
ger. The letter E means “times 10 to the power of.” The digits that fol- low the E compose the exponent. The second expression represents 3.5562×10−99. This is an extremely small positive rational number.
Quiz
This is an “open book” quiz. You may refer to the text in this chapter. A good score is 8 correct. Answers are in the back of the book.
1. Consider two numbers pand q, such that p=q+ 10,000. By how many orders of magnitude do pandqdiffer?
(a) 10,000. (b) 100. (c) 4.
(d) More information is necessary to answer this.
2. How many significant figures does the numeric expression 3,700.00 have? (a) 6.
(b) 4. (c) 3. (d) 2.
3. In the coordinate system shown by Fig. 3-4, the horizontal scale (a) is linear.
(b) spans 1 order of magnitude. (c) spans 10 orders of magnitude. (d) spans 100 orders of magnitude.
4. In the coordinate system shown in Fig. 3-4, the largest value that can be plotted on the yaxis, as it is drawn, differs from the smallest value that can be plotted on that same axis, as it is drawn, by a factor
(a) that can’t be determined without more information. (b) of 102.
(c) of 103.
(d) of 104.
5. Suppose you have a positive real number. Call it x. First, you square x. Then, you square the result of that. Next, you cube the result of that. Finally, you take the fourth power of the result of that. What is the final value in terms of x?
(a) 1 (b) x10
(c) x48
6. Suppose you have a positive real number. Call it y. First, you square y. Then, you square the result of that. Next, you cube the result of that. Finally, you take the 1⁄
3power of the result of that. What is the final value
in terms of y? (a) 1
(b) y2
(c) y4
(d) There is no way to tell without knowing the exact value of y.
7. Consider two numbers pandq, such that p=q/100. By how many orders of magnitude do pandqdiffer?
(a) 100 (b) 10 (c) 2
(d) None of the above
8. Suppose you have a positive real number, expressed to 4 significant fig- ures. Call it z. First, you take the fourth power of z. Then, you divide the result by the square root of z. Finally, you square the result of that. When
0.01 0.1 1.0 10 10 30 70 100 Quantityx Quantity y
you get the final answer, how many significant figures can you legiti- mately claim?
(a) 8. (b) 6. (c) 4.
(d) The value is theoretically exact, so you can claim as many signifi- cant figures as you want.
9. Suppose you have a positive real number, expressed to 4 significant figures. Call it w. First, you take the fourth power of w. Then, you divide the result by the square root of w. Finally, you take the zeroth power of that. When you get the final answer, how many significant figures can you legitimately claim?
(a) 8. (b) 6. (c) 4.
(d) The value is theoretically exact, so you can claim as many signifi- cant figures as you want.
10. Imagine a variable x that starts out as a large number negatively (for example, x= −1000), and increases in value, passing through x = −100, thenx= −10, then x= −1, then x= −0.1, then x= −0.01, and so on, closer and closer to 0. Suppose that the value of x approaches 0, and gets arbitrarily close without limit, yet never reaches 0 or becomes positive. If x is expressed in scientific notation, how does the power of 10 change as xvaries in this way?
(a) It gets larger and larger, and is always positive. (b) It gets smaller and smaller, and is always negative.
(c) It starts out positive, gets smaller and smaller, passes through 0, and then becomes larger and larger negatively.
(d) It starts out negative, gets larger and larger (that is, smaller and smaller negatively), passes through 0, and then becomes larger and larger positively.