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how do you prove that sqrt(2) + sqrt(5) is irrational...
i did get it down to 7+2root10 = m^2/n^2 where i assumed root 2 + root 5 = m/n...and before that i proved root 10 to be irrational...so i am kinda stuck on the 7+2root10 = m^2/n^2 bit...how does that mean anything...matt grime said:by contradiction, suppose r=sqrt(2)+sqrt(5) and r is rational. play around with that and see what you can show this implies.
for √2 + √3, let it equal a...
a^2 = 5 + 2√6
a^2 - 5 = 2√6
a^4 - 10a^2 + 25 = 24
a^4 - 10a^2 + 1 = 0
The minimum polynomial of a is x^4 - 10x^2 + 1. (I haven't proven it actually is the minimum, but it will still suffice for this method of proof)
Now, the only possible rational roots of this are 1 and -1, and neither of these is √2 + √3, so it's irrational
this is where i got up to...let root 2 + root 5 = m/n...i.e. assume they are rational...and we have already proved that root 10 is irrational...
squaring both sides:
2+5+2.root2.root5 = m2/n2
or 7+2.root10=m2/n2
so we have irrational = "(m2/n2 - 7)/2"root 10 = (m2/n2 - 7)/2
HallsofIvy said:Yes, that is nice.
Irrationality is a mathematical concept that refers to a number that cannot be expressed as a ratio of two integers. These numbers are typically represented by decimal expansions that do not terminate or repeat.
There are several methods for proving irrationality, including the proof by contradiction, the proof by the infinitude of primes, and the proof by continued fractions. These methods involve showing that a number cannot be expressed as a ratio of two integers, thus proving its irrationality.
No, not every number can be proven to be irrational. In fact, the majority of numbers are irrational. This is because there are infinitely more irrational numbers than rational numbers. However, some numbers, such as whole numbers and terminating decimals, can be proven to be rational.
Some examples of famous irrational numbers include pi (3.1415926...), the golden ratio (1.6180339...), and the square root of 2 (1.4142135...). These numbers have infinite decimal expansions that do not terminate or repeat, making them irrational.
Proving irrationality is important in mathematics because it helps us understand and classify different types of numbers. It also helps us solve problems and make connections between different mathematical concepts. Additionally, irrational numbers have many real-world applications, such as in geometry, physics, and finance.