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Every rational number is algebraic (since $\frac{a}{b}$ is the root of $bx-a$); if you can prove a number is not algebraic, then it must be irrational. For example, one can prove that $e$ and that $\pi$ are transcendental, but showing that they cannot be roots of any polynomial with integer coefficients; in particular, they cannot be rational ...
Proof: Assume that $\sqrt{2}$ is rational, then $\sqrt{2}= {p\over q}$ for some integer $p$ and $q$ with $q≠0$ and gcd($p$,$q$)=$1$. (2 things to note here: First, the only thing we assumed is that $\sqrt{2}$ is rational. Everything else above follows directly from that assumption and the definition of rational.
Learn the difference between rational and irrational numbers, learn how to identify them, and discover why some of the most famous numbers in mathematics, like Pi and e, are actually irrational. Did you know that there's always an irrational number between any two rational numbers?
In this proof we want to show that √2 is irrational so we assume the opposite, that it is rational, which means we can write √2 = a/b. Now we know from the discussion above that any rational number that is not in co-prime form can be reduced to co-prime form, right?
Rational numbers are all numbers that can be written as the ratio (or fraction) of 2 integers. This is the basic definition of a rational number. Here are examples of rational numbers:-- All integers. Numbers like 0, 1, 2, 3, 4, .. etc. And like -1, -2, -3, -4, ... etc. -- All terminating decimals. For example: 0.25; 5.142; etc.
Irrational Numbers. An Irrational Number is a real number that cannot be written as a simple fraction: 1.5 is rational, but π is irrational. Irrational means not Rational (no ratio) Let's look at what makes a number rational or irrational ... Rational Numbers. A Rational Number can be written as a Ratio of two integers (ie a simple fraction).
Euclid proved that √2 (the square root of 2) is an irrational number. He used a proof by contradiction. First Euclid assumed √2 was a rational number. A rational number is a number that can be in the form p/q where p and q are integers and q is not zero.
