Acc. Geom/Adv. Algebra Name KEY .
“Proofs” of Properties of Numbers Period______Date______
Statement: The product of two irrational numbers is irrational.
Example: Both 3 and 7 are irrational, so their product, 3 7 21 which is
also irrational.
Counterexample: Both 3 and 2 3 are irrational, but their product 3 2 3 =
23 = 6, which is NOT irrational.
Statement: The quotient of two natural numbers is not natural.
Example: 6 and 8 are natural numbers, but 6/8 = ¾, which is not a natural number.
Counterexample: 6 and 2 are natural numbers, but 6/2 = 3 which is a natural
number.
Statement: The area of a 30˚-60˚=90˚ triangle is irrational.
Example: If the shorter leg of the triangle is 2, the longer leg is 2 3 , so the area
is A = ½ baseheight = ½ (2)(2 3 ) = 2 3 which is irrational.
Counterexample: (Thought process: In general, the area of a 30˚-60˚-90˚
triangle is ½ s2 3, where s is the length of the shorter leg. If this is to be a
rational number, then s2could equal 3. That would mean that s2 = 31/2 so s =31/4.)
If the short leg has length 31/4, the longer leg would have length 31/4 3 = 31/431/2
= 33/4. The area would be A = ½ baseheight = ½ 31/433/4 = ½ 34/4 = 3/2 which is
rational.
Statement: The perimeter of a 45˚-45˚-90˚ triangle is rational.
Example: Will be done Friday in class…
Counterexample: If each leg of the triangle has length 2, then the hypotenuse has length 2 2 and the perimeter will be 2 + 2 + 2 2 = 4 + 2 2 which is irrational.
Statement: The area of a circle is irrational.
Example: If the radius of the circle is 2, the area = πr2 = π22 = 4π which is
irrational.
Counterexample: (Thought process: If πr2 is to be rational, the π must “go away”, and the only way that can happen is if r2 will multiply by π to be 1 (or some other rational number.) If πr2 = 1, then r2 = 1/π and so r = 1/.) If r = 1/, A = πr2 = π( 1/)2 = π(1/π) = 1, which is a rational number.