By Cahen M., Gutt S., Rawnsley J.
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Suppose after a party we have 43 unopened bottles of beer. We’d like to store them in containers that hold 12 bottles each. How many containers will we need? 58333. So we need 3 boxes and another 7 twelfths of a box. Obviously we really need 4 boxes – at least one will have some unused space in it. In this sort of situation we’re dealing with the ceiling function. Given a real number, the ceiling function rounds it up to the next integer. Both of these functions are denoted using symbols that look very much like absolute value bars.
Here are two such false conjectures regarding prime numbers. 2. 1: The first three stages in the sieve of Eratosthenes. What is the smallest composite number that hasn’t been crossed off? Conjecture 1. Whenever p is a prime number, 2p − 1 is also a prime. Conjecture 2. The polynomial x2 − 31x + 257 evaluates to a prime number whenever x is a natural number. In the exercises for this section, you will be asked to explore these statements further. Prime numbers act as multiplicative building blocks for the rest of the integers.
12] While the study of algorithms is more properly a subject within Computer Science, a student of Mathematics can derive considerable benefit from it. There is a big difference between an algorithm description intended for human consumption and one meant for a computer7 . The two favored humanreadable forms for describing algorithms are pseudocode and flowcharts. The former is text-based and the latter is visual. There are many different modules from which one can build algorithmic structures: for-next loops, do-while loops, if-then statements, goto statements, switch-case structures, etc.