0
If a and b are rational, with a < b, then a + (b-a) [sqrt(2)/ 2] is an irrational number between a and b.
This number is between a and b because sqrt(2)/2 is less than one and positive, so that a < a + (b-a) [sqrt(2)/3] < a + (b-a) [1] = b.
To prove that a + (b-a) [sqrt(2)/2] is not rational, suppose that
a + (b-a) [sqrt(2)/2] = p/q where p and q are integers.
Then, sqrt(2) = ( p/q -a ) 2/(b-a) which is rational since the rationals are a field, closed under arithmetical operation, but sqrt(2) not rational (Look up the elementary proof if you do not know it.)
Still curious? Ask our experts.
Chat with our AI personalities
Maxine
Vivi
ReneAdd your answer:
Earn +20 pts
How do you insert rational numbers between two rational numbers?
Find the arithmetic average of the two rational numbers. It will be a rational number and will be between the two numbers.
How do you Find two rational and irrational nos. between 2.1 and 2.11?
For two rational numbers select any terminating or repeating decimal number which starts with 2.10 and for irrational numbers you require a non-terminating, non-repeating decimal which also starts with 2.10.
Can the rational zero test be used to find irrational roots as well as rational roots?
Rational zero test cannot be used to find irrational roots as well as rational roots.
Find rational numbers between 0 and -1?
Oh, dude, finding rational numbers between 0 and -1 is like trying to find a unicorn at a zoo. It's just not gonna happen. Rational numbers are all about fractions, and you can't have a fraction where the numerator is smaller than the denominator. So, in this case, there are no rational numbers between 0 and -1. It's a mathematical dead end, my friend.
How do you find a rational number between two rational numbers?
Suppose the two rational numbers are x and y.Then (ax + by)/(a+b) where a and b are any positive numbers will be a number between x and y.
