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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 The highest wind speed in a category 1 hurricane is 95 miles per hour The highest wind speed in a category 4 hurricane is 155 miles per hour What is the ratio of these wind speeds, expressed in lowest terms between two integers 2 On the absolute temperature scale, the coldest possible temperature is 0, defined in units called kelvins On this scale, pure water at sea level freezes at 273 kelvins and boils at 373 kelvins (to the nearest kelvin) Based on this information, what is the ratio of the absolute temperature of the boiling point to the absolute temperature of the freezing point, expressed in lowest terms between two integers 3 Is the fraction 231/230 in lowest terms If not, reduce it to lowest terms 4 Is the fraction 154/165 in lowest terms If not, reduce it to lowest terms 5 If two fractions are in lowest terms and they are multiplied by each other, is the product always in lowest terms If so, prove it If not, provide an example If the product is sometimes in lowest terms but not always, provide examples of both situations 6 If two fractions are in lowest terms and one is divided by the other, is the quotient always in lowest terms If so, prove it If not, provide an example If the quotient is sometimes in lowest terms but not always, provide examples of both situations 7 In one of the challenge problems, you found a general expression for [(a /b)/(c /d )]/(e /f ) when a, b, c, d, e, and f are all nonzero integers You divided a /b by c /d, and then divided the result by e /f Now find a general expression for (a /b)/[(c /d )/(e /f )] Once you ve done this, you ll know what happens when you divide c /d by e /f first, and then divide a /b by the result Compare this with the formula you got when you solved the challenge problem 8 Imagine that you have a fraction of the form a /b and another of the form c /d, where a and c are integers, and b and d are positive integers Show that the commutative law works for multiplication of these fractions, based on your knowledge of the commutative law for integers 9 Imagine that you have fraction of the form a /b, another of the form c /d, and a third of the form e /f, where a, c, and e are integers, and b, d, and f are positive integers Show that the associative law works for multiplication of these fractions, based on your knowledge of the associative law for integers 10 Give at least one example of a situation in which the following equation is true: (a /b)/(c /d ) = (c /d )/(a /b) where a, b, c, and d are integers Don t use any of the trivial cases where all four of the integers have absolute values of 1.

The explore procedure visits only the portion of the graph reachable from its starting point To examine the rest of the graph, we need to restart the procedure elsewhere, at some vertex that has not yet been visited The algorithm of Figure 35, called depth- rst search (DFS), does this repeatedly until the entire graph has been traversed The rst step in analyzing the running time of DFS is to observe that each vertex is explore d just once, thanks to the visited array (the chalk marks) During the exploration of a vertex, there are the following steps: 1 Some xed amount of work marking the spot as visited, and the pre/postvisit 2 A loop in which adjacent edges are scanned, to see if they lead somewhere new

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Now that you know how a rational number can be expressed as a ratio of two integers, let s look at the other common way these numbers are symbolized Since the middle of the twentieth century, calculators and computers have replaced manual methods of calculation These machines use decimal fractions

AAII s members were asked to finish this sentence: I feel that the direction of the stock market over the next six months will be: Bearish, Neutral, or Bullish

A positive-integer power is a quantity multiplied by itself a certain number of times If a nonzero quantity is divided by itself once, it is said to be raised to the zeroth power, and the result is always 1 A negative-integer power is a nonzero quantity divided by itself more than once Powers are denoted by exponents Decimal notation is based on integer powers of 10 The number 10 is called the exponential base It can also be called simply the base or the radix

As with many of our graph algorithms, this one applies to both undirected and directed graphs In such cases, we adopt the directed notation for edges, (x, y) If the graph is undirected, then each of its edges should be thought of as existing in both directions: (x, y) and (y, x)

Orders of magnitude Figure 7-1 is a number line showing the powers of 10, from 105 (the largest number in the illustration) down to 10 5 (the smallest) Each multiple of 10 is called an order of magnitude For example, 103 is one order of magnitude larger than 102, and 10 2 is three orders of magnitude larger than 10 5 This number line differs from the ones you ve seen so far All the values here are positive As you go upward on the line, the numerical value increases faster and faster, so you race off toward infinity more rapidly than you do on a conventional number line As you go downward, the value decreases at a slower and slower rate, closing in on 0 but never quite getting there If you expand on this idea, you can build any positive rational number by taking singledigit multiples of powers of 10, and adding them up Every positive number in this form has its negative twin To show the negative rational numbers, you can make up a separate number line for them In order to account for 0, you can give it a special point that isn t on either line Figure 7-2 portrays all the rational numbers using this system Infinity and negative infinity are not entitled to points here, because neither of them is a rational number!

331%

Figure 35 Depth- rst search procedure dfs(G) for all v V : visited(v) = false for all v V : if not visited(v):

positive rational number line, showing values from 100,000 down to 1/100,000 The powers of 10 range from 5 down to 5

Continues forever! 10

explore(v)

100,000

324%

10,000

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