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a = N 3 + 2,50 3.1462643699419723423291350657155704455124771291873 b = N 3 2,50 0.31783724519578224472575761729617428837313337843343 a*b 1.0000000000000000000000000000000000000000000000000 n! 2.8 The binomial coefficient C (n, k ) = k !(n k )! . Use this definition to compute C(10,4).
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2.9 A fair die has six faces, numbered 1 through 6, and each occurs with equal probability. Simulate four tosses of a fair die.
SOLUTION
(Your answers will be different from those shown here.) Random[Integer, {1, 6}] Random[Integer, {1, 6}] Random[Integer, {1, 6}]
Basic Concepts
Random[Integer, {1, 6}] 6 1 5 3
2.10 Find a 15 significant digit pseudorandom real number between and 2 .
SOLUTION
(Your answer will be different from that shown here.) Random[Real,{ , 2 }, 15] 4.13129131207734
2.11 What is the 27th Fibonacci number
SOLUTION
Fibonacci[27] 196 418
2.12 Show that there is no prime between 157 and 163.
SOLUTION
We determine this by experimentation.
Prime[37] 157 Prime[38] 163
Since 157 and 163 are consecutive primes, there is no prime between them.
2.13 What is the integer closest to 159
SOLUTION
Round[Sqrt[159]] 13
159 //Round
2.14 Between what two consecutive integers does ( 2 + 1)5 lie
SOLUTION
Floor[( 2 +1)5] 151 729 Ceiling[( 2 +1)5 ] 151 730 The number ( 2 + 1)5 lies between 151,729 and 151,730.
2.15 Compute the value of x x first using x = 17 and then using x = .
SOLUTION
x=17; Ceiling[x] Floor[x] 0 x = Pi; Ceiling[x] Floor[x] 1
x x always equals 0 when x is an integer and 1 when x is not an integer.
Basic Concepts
2.16 What are the greatest common divisor and least common multiple of 5355 and 40425
SOLUTION
GCD[5355, 40425] 105 LCM[5355, 40425] 2 061 675
2.17 Show that 15, 16, and 30 are relatively prime (integers are relatively prime if they have no common factor other than 1).
SOLUTION
GCD[15, 16, 30] 1 Since their GCD = 1, their only common factor is 1. Therefore, they are relatively prime.
2.18 A theorem from number theory says that the product of the GCD and LCM of two numbers is always equal to the product of the numbers. Verify this using the numbers 74613 and 85085.
SOLUTION
a = 74 613; b = 85 085; GCD[a, b]* LCM[a, b] 6 348 447 105 a*b 6 348 447 105 Obviously, the products are identical.
2.19 Show that 156,875,438,767 is not prime and factor.
SOLUTION
PrimeQ[156 875 438 767] False FactorInteger[156 875 438 767] {{53,1},{2 959 913 939,1}} 156,875,438,767 is equal to the product of primes 53 and 2,959,913,939.
2.20 How long did it take Mathematica to factor 156,875,438,767 in the previous problem
SOLUTION
Timing[FactorInteger[156 875 438 767]] {0.011 ,{{53, 1}, {2 959 913 939, 1}}} It took approximately 0.011 seconds. (This time will vary from computer to computer.)
2.21 Compute the natural logarithm of e5.
SOLUTION
Log[ 5] or Log[E^5] or Log[Exp[5]] 5
Basic Concepts
2.22 Compute the common logarithm (base 10) of e5. What is its numerical approximation
SOLUTION
Log[10, 5] or Log[10, E^5] or Log[10, Exp[5]] 5 Log[10] % //N 2.17147
2.23 If Jacob starts with one cent and his money doubles every day, how much money will he have, to the penny, after 30 days
SOLUTION
N [230 /100] 1.07374 107 If we want to get the amount to the penny, we will need 10 significant digits. amount = N[230 /100, 10] 1.073741824 107 To see this in a more traditional format, the function AccountingForm can be used. AccountingForm[amount] 10737418.24 We can group the digits into blocks of 3 and separate them with commas using the option DigitBlock AccountingForm[amount, DigitBlock 3] 10,737,418.24
2.24 What is the exact value of sin 15 Compute a 20 decimal place approximation.
SOLUTION
Sin[15 Degree] 1 + 3 2 2 N[%, 20]
Sin[15 ]
0.25881904510252076235
2.25 Select a random number, x, between 0 and 1 and compute sin2 x + cos2 x.
SOLUTION (Your value of x will be different from that shown here.)
x = Random[ ] 0.427468 Sin[x]2 + Cos[x]2 1.
Recall from trigonometry that sin2 x + cos2 x = 1 for all x.
2.26 Find a number between /2 and /2 whose sin is 1/2.
SOLUTION
ArcSin[1/2] 6
Basic Concepts
2.27 Select a random number, x, between 0 and 1 and compute cosh2 x sinh2 x.
SOLUTION (Your value of x will be different from that shown here.)
x = Random[] 0.991288 Cosh[x]2 Sinh[x]2 1.
Hyperbolic functions have properties similar to trigonometric functions: cosh2 x sinh2 x = 1 for all x.
2.28 Obtain an alternate representation of tanh(ln x).
SOLUTION
Tanh[Log[x]] //TraditionalForm x2 1 x2 + 1
2.29 Approximately how many radians are there in one degree
Approximately how many degrees are there in one radian
SOLUTION
N[Degree]
N[1/Degree] 57.2958
Degree is a Mathematica constant which represents the number of radians in one degree. 1/Degree represents the number of degrees in one radian.
2.30 How much is + 100,000
SOLUTION
+ 100 000
2.31 What is the square root of the complex number 3+4i
SOLUTION
3+4 2+
Sqrt[3 + 4 I]
n 2.32 The number of permutations of n objects taken k at a time is P(n, k ) = (n !k )! . How many permutations of 20 objects taken 10 at a time are there
SOLUTION
n = 20; k = 10; n!/(n k)! or Factorial[n]/Factorial[n k] 670 442 572 800
2.33 Between what two consecutive integers does the natural logarithm of 100,000 lie
SOLUTION
Floor[Log[100 000]] 11 Ceiling[Log[100 000]] 12 ln 100,000 lies between 11 and 12.
Basic Concepts
2.34 What is the quotient and remainder if 62,173,467 is divided by 9,542
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