Spooky Action At A Distance

By the editors

Of course you've graduated from high school or college or whatever, and now when you see letters arranged in a semantically correct sequence in some variant of the English langauge you can read the words they spell, hopefully without moving your lips. But can you grasp the semantic content of the symbolic sequence? That is, assuming the author had something to communicate and the facility to string together words with senamtic intent, can you ascertain that intention? Note that often it seems many journalists are unable to work either side of this process effectively, so considering yourself some kind of writer or intellectual may not be of any significance here. (An interesting example was a news person on the Lehrer News Hour>/i> recently confusing the string rate with the sense implied by the string total.)

At any rate, read the following simple paragraphs written in common English and then answer the questions. It should test your reading ability at about a fifth grade level (in a developed foreign country where English is a second language, like Germany or France).

In the following, assume that the letter n represents an even integer greater than zero with a rational number as a square root. (A rational number is a fraction; that is to say, a ratio of integers with the integer on the bottom not zero. However, by the assumption on n we can restrict our attention to ratios of integers greater than zero. A square root of a positive number represented by a is a number represented by b with the property that the number represented by b multiplied by itself is the number represented by a; assume that the number represented by b is positive. We write b*b to represent multiplication of b by itself and in general, the expression a*b is written to indicate the multiplication of the numbers represented by a and b. We use * to indicate multiplication so as to avoid such ambiguity as "as", for example. By an even integer is meant a number a = 2*b where b represents an integer. The symbol = means the two sides represent the same number.)

By assumption n can be written as n = (c/d)², where the letters c and d represent integers that are relatively prime and greater than zero, the / indicates simple division, while for an integer (or any real number a), a² represents the operation a*a. (For two integers to be relatively prime means that they have no common factors other than the integer 1. In other words, the only integer that divides both of the integers represented by c and d is 1.)

That the integers represented by c and d are relatively prime implies that at least one of the integers represented by c and d cannot be an even integer. Since n represents an even integer, it can be written as n = 2*x where x represents an integer greater than zero. By simple substitution of expressions as if they were the numbers they represent and then multiplication by the number represented by d, it can be determined that c² = 2*x*(d²). Therefore, c² is even. Because an integer which is the product of an integer with itself is even if and only if it's root is even, we know that for some integer y greater than zero, c = 2*y. (A number which is the product of an integer with itself, that is, of the form a*a where a represents an integer, is called a perfect square.)

Simple arithmetic with the integers represented by the letters c, d, x and y leads us to conclude that 4*(y²) = c² = 2*x*(d²). That is, 2*(y²) = x*(d²). Conclude that the integer represented by x*(d²) is even. But the integer represented by d cannot be even, since the integers represented by c and d are relatively prime, so the integer represented by x must be even. Hence there must be an integer represented by the symbol z so that x = 2*z. This means n =2*x = 4*z. That is, the integer represented by n must be a muliple of 4 by an integer, or n = 4*a where a represents an integer.

In the following questions you can only draw conclusions based on the information provided above. No other source is allowed, including your own calculations or prior knowledge. There can be more than one correct answer to each question, and it is required that all correct answers be chosen.

1) According to the discussion above, the square root of 2 is
a) the ratio of two integers
b) a fraction
c) an integer
d) not the ratio of two integers
e) not a real number
f) an even number
g) not a fraction

2) According to the discussion above, 81 is
a) a perfect square
b) an even number
c) not a rational number
d) a rational number
e) not a square root

3) According to the discussion above, the square root of 3 is a fraction.
a) true
b) false
c) we can't say

4) According to the discussion above, the square root of 8 is a rational number.
a) true
b) false
c) we can't say

5) According to the discussion above, the ratio of two positive integers is a fraction.
a) true
b) false
c) we can't say

6) According to the discussion above, if a represents an even integer and a = 4*b with b representing an integer, the square root of a is a fraction.
a) true
b) false
c) we can't say

7) According to the discussion above, if a represents an even integer and a ≠ 2*b with b representing an even integer, then the square root of a cannot be a fraction.
a) true
b) false
c) we can't say

8) According to the discussion above, if a represents an integer and a ≠ 4*b with b representing an integer, then the square root of a cannot be a fraction.
a) true
b) false
c) we can't say

9) According to the discussion above, if a represents an integer, then in order for the square root of a to be a fraction, a = 2*b with b representing an integer.
a) true
b) false
c) we can't say

10) According to the discussion above, 36 = 4*9 has a rational square root.
a) true
b) false
c) we can't say

11) According to the discussion above, 12 and 15 are relatively prime.
a) true
b) false
c) we can't say

12) According to the discussion above, ½ and 2 are relatively prime.
a) true
b) false
c) we can't say

13) According to the discussion above, if a represents an even integer and a ≠ 4*b where b represents an integer, then the square root of a cannot be a fraction.
a) true
b) false
c) we can't say