Download MATH 5 ASSIGNMENT 16: SQUARE ROOT OF 2 M • We talked

MATH 5
ASSIGNMENT 16: SQUARE ROOT OF 2
MATERIAL COVERED TODAY
• We talked about √
Pythagorean theorem and Egyptian triangle (3, 4, 5).
• We showed that 2 is not a rational number (that is it cannot be written as a fraction).
√
Let us assume that 2 = pq where p and q are some whole numbers and the fraction pq cannot be
2
√
2
simplified anymore (for example instead of 46 we will always write 23 ). We write ( 2)2 = pq or 2 = pq2 .
We have
p2 = 2q 2
and one can see that p must be an even number. Then it can be written as p = 2m, where m is some
whole number. We have (2m)2 = 2q 2 or q 2 = 2m2 and one conclude the same way that q is also an even
number. But if this is true the fraction pq can be simplified by dividing both p and q by 2. But we assumed
√
that pq can not be simplified. We have obtained the contradiction which proves that 2 cannot be written as
p
q with whole p and q.
√
• We found an approximate value of 2 using a special trick (see the problem 5 below).
HOMEWORK
√
√ 2 √ 4 √ 3 √
4
6
1. Simplify: ( 17) , ( 13) , ( 11) , 2 3 , 24 35 .
√
2. Can you show that 3 is NOT a rational number, that is it cannot be written as
numbers p and q.
p
q
with some whole
3. Can one cut the square with the side of 1 m from the circle of the diameter 1.4 m?
4. The side of an equilateral triangle is 1 m. Find its height and the area.
*5. Take some positive number x < 100 and using calculator (or computer) calculate the number x2 + x1 . Call
the result x and repeat the same calculation with the new x. Do it 10 times. Then take the result and
square it. What did you get? Try to do the same thing starting with different numbers. Is it surprising?
6. How many behemoths can take one truck with maximal load of 5 tonnes (5000 kg) if the weight of each
behemoth is 1500 kg? How many crocodiles can take the same truck if the weight of each crocodile is
175 kg?