You take the sum of an irrational and a rational number-- and we'll see this later on. It never terminates. And we call these numbers irrational numbers. And in a future video, we'll prove that you give me two rational numbers-- rational 1, rational there's going to be at least one irrational number between those, which is a neat result, because irrational numbers seem to be exotic.
So there's a lot, a lot, a lot of irrational numbers out there.
Since the set of rational numbers is countableand the set of real numbers is uncountablealmost all real numbers are irrational. And it turns out-- as you can imagine-- that actually some of the most famous numbers in all of mathematics are not rational.
Rational numbers definition with example
The sum of an irrational and a rational is going to be irrational. The product of an irrational and a rational is going to be irrational. So these things that really just pop out of nature, many of these numbers are irrational. So negative 7 is definitely a rational number. And there's many, many, many other examples of this. Finite extensions of Q are called algebraic number fields , and the algebraic closure of Q is the field of algebraic numbers. How can we represent that as the ratio of two integers? Or you could say, hey, 3.
Square root of 2, irrational number. It comes out of complex analysis. Phi, the golden ratio, irrational number.
Rational number explain
Well, we could go on and on. What is left? Or you could say, hey, 3. I'm giving you multiple examples of how this can be represented as the ratio of two integers. Rational numbers together with addition and multiplication form a field which contains the integers and is contained in any field containing the integers. You've included all of finite non-repeating decimals, and you've also included repeating decimals. And I could go on, and on, and on, and on. You've included all of the integers. Irrational numbers Video transcript So let's talk a little bit about rational numbers.
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over.
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