# Irrational Numbers

## An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. While their existence was once kept secret from the public for philosophical reasons, they are now well accepted, yet still surprisingly hard to prove on an individual basis. Please post all questions about irrational numbers, including the famous examples of π, e, and √2, into this category.

### What is the letter symbol for irrational numbers? Q with an apostrophe. (Q') H with a bolded side. Meaning any number that cannot be written as a fraction, decimal that does not repeat or terminate.

### Give a proof that the square root of 7 is an irrational number? Proof by contradiction: suppose that root 7 (I'll write sqrt(7)) is a rational number, then we can write sqrt(7)=a/b where a and b are integers in their lowest form (ie they are fully cancelled). Then square both sides, you get 7=(a^2)/(b^2) rearranging gives (a^2)=7(b^2). Now consider the prime factors of a and b. Their squares have an even number of prime factors (eg. every prime factor of a is there twice in a squared). So a^2 and b^2 have an even number of prime factors. But 7(b^2) then has an odd number of prime factors. But a^2 can't have an odd and an even number of prime factors by unique factorisation. Contradiction X So root 7 is irrational.

### What is an irrational number that is also rational? Numbers are either irrational (like the square root of 2 or pi) or rational (can be stated as a fraction using whole numbers). Irrational numbers are never rational.

### Is the set of all irrational number countable? No, it is uncountable. The set of real numbers is uncountable and the set of rational numbers is countable, since the set of real numbers is simply the union of both, it follows that the set of irrational numbers must also be uncountable. (The union of two countable sets is countable.)

### What are the subsets of integers? There are an infinite number of subsets. The empty set, Even numbers Odd numbers Multiples of 3 Multiples of 4, etc Integers from 1 to 31 (days of the month) Integers from 0 to 23 (hour display on some digital clocks) {1}, {1,4,456,-5}, (1,2,3,4,5,6,7,3492} and so on, and on, and on. There are an infinite number of possible subsets. Odd numbers, even numbers, the age (in whole years) of each member of my family, my friends' telephone numbers, prime numbers, integers between 70 and 83 are all examples of subsets.

### Is the square root of 50 a irrational number? 7.071678118....... rounded to 7.1 ***** To a whole lot of places, it's actually 7.0710678118654752440084436210485... But no finite number of places can be enough to establish its irrationality. Certainly, √50 is indeed irrational. Here is how we can see that it is so: First of all, we note that √50 = 5√2. To show that the above is irrational, we need to show two things: (1) that √2 is irrational; and (2) that the product of a integer and an irrational is irrational. We show (1), above, by reductio ad absurdum: We assume the opposite of that which we wish to prove and show that it entails a contradiction. Suppose that √2 is rational. Then it must equal the quotient p/q, where p and q are two integers that are relatively prime. In particular, not both, p and q may be even numbers. (√2 might also equal some p'/q', where p' and q' have a common divisor greater than 1; but, in that case, the fraction could be reduced to p/q. Thus, we can assume, without loss of generality, that p and q are relatively prime; that is, that p/q is irreducible.) Before continuing our proof, we shall need to establish the following: The square of every even integer is an even integer; and the square of every odd integer is an odd integer. Let n be even; thus, n = 2k, where k is some integer. Then, n2 = 4k2; thus, n2 is divisible by 4 and, therefore, it is also divisible by 2. Let m be odd; thus, m = 2j + 1. Then m2 = (2j + 1)2 = 4j2 + 4j + 1 = 2(2j2 + 2j) + 1, which is evidently odd. We may conclude, from the above, that, if a perfect square is an even integer, then its square root must also be even. Now, we may continue the main proof: From √2 = p/q, which we assumed above, p and q being relatively prime, we deduce the following: p2 = 2q2; therefore, p2 is even; therefore, p is even. Because p is even, p = 2r, for some integer r. Then, p2 = 4r2 = 2q2; thus, q2 = 2r2, and q is similarly shown to be even. P and q, therefore can not be relatively prime. But, wait! Our assumption was that p and q are relatively prime and can not both be even; and, so, we have the contradiction we sought. Therefore, (1) is proved, and there exist no integers p and q, such that p/q = √2, and p/q is an irreducible fraction. In other words, √2 must, after all, be irrational. Now, we proceed to show (2), that the product of an integer and an irrational is irrational; it will suffice to do this by means of the example in hand: namely, that, if √2 is irrational, then √50 = 5√2 must also be irrational. First, we know that the sum of two integers is also an integer; likewise, by induction, the product of two integers is an integer. From this, we can show that the product of two rationals is also a rational: Let the two rationals be a/b and c/d, where a, b, c, and d are all integers. Then, their product is ac/bd, and, because ac and bd are integers, ac/bd must, by definition, be rational. Multiplying both sides by 1/5, we obtain (1/5)(√50) = √2. Now, 1/5 is rational; then, if √50 is rational, then √2, being the product of two rationals, must also be rational. But we have shown, in (1) above, that √2 is not rational. We conclude that √50 can not be rational, either; therefore, it is irrational, which is what we set out to prove.

### Is every square root an irrational number? No. Sqrt(4) = 2, sqrt(0.04) = 0.2 are examples where the square roots are rational. Sqrt(3) is irrational Sqrt(-3) is neither rational nor irrational but imaginary.

### Is the sum of three irrational numbers an irrational number? Yes, but not always. An easy example is sqrt(2) + sqrt(2) + sqrt(2) = 3sqrt(2), an irrational number. An easy counterexample is 2sqrt(2) + -sqrt(2) + -sqrt(2) = 0, which is rational.

### Prove that square root of 6 is irrational number? If a/b=sqrt(6), then a2=6b2 On the other hand, given integers ''a'' and ''b'', because the valuation (i.e., highest power of 2 dividing a number) of 6b2 is odd, while the valuation of a2 is even, they must be distinct integers. Contradiction.

### What is the symbol for irrational number? There is no special symbol for irrational numbers in general. There are symbols for some special numbers, such as pi and e.

### Is a number multiplied by pi irrational? Maybe. It depends on the number that you are multiplying. Pi multiplied by (1 / 4pi) = 1/4. A quarter is definitely rational.

### Can the quotient of two irrational numbers be rational? Yes. 2*pi is irrational, pi is irrational, but their quotient is 2pi/pi = 2: not only rational, but integer.

### Who is the founder of irrational numbers? The history of this is confused. The first irrational number found was almost certainly root 2. The man who proved this was probably Hippasus. Hippasus died by drowning, and some stories say that Pythagoras drowned him for proving the existence of an irational number which he thought was impossible. Others say that Pythagoras ordered the drowning of Hippasus for revealing the existence of an irrational number which should be kept fron public knowledge.

### What is the difference between rational numbers and irrational numbers? An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be. A rational number is defined to be a number that can be expressed as the ratio of two integers. An irrational number is any real number that is not rational. A rational number is a number that can be expressed as a fraction. An irrational number is one that can not. Some examples of rational numbers would be 5, 1.234, 5/3, or -3 Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3. A rational number is a number that either has a finite end or a repeating end, such as .35 or 1/9 (which is .1111111 repeating). An irrational number has an infinite set of numbers after the decimal that never repeat, such a the square root of 2 or pi. A rational number is one that can be expressed as a ratio of two integers, x and y (y not 0). An irrational number is one that cannot be expressed in such a form. In terms of decimal numbers, a rational number has a decimal representation that is terminating or [infinitely] recurring. The decimal representation for an irrational is neither terminating nor recurring. (Recurring decimals are also known as repeating decimals.) A rational number is a number that can be expressed as a fraction. An irrational number is one that can not. Some examples of rational numbers would be 5, 1.234, 5/3, or -3 Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3. An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be. A rational number can be represented by a ratio of whole numbers. An irrational number cannot. There are many more irrational numbers than there are rational numbers Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction. A rational number can be expressed as a fraction, with integers in the numerator and the denominator. An irrational number can't be expressed in that way. Examples of irrational numbers are most square roots, cubic roots, etc., the number pi, and the number e. A rational number can always be written as a fraction with whole numbers on the top and bottom. An irrational number can't. A rational number can always be written as a fraction with whole numbers on top and bottom. An irrational number can't. Any number that you can completely write down, with digits and a decimal point or a fraction bar if you need them, is a rational number. A rational number can be expressed as a fraction whereas an irrational can not be expressed as a fraction. Just look at the definition of a rational number. A rational number is one that can be expressed as a fraction, with integers (whole numbers) in the numerator and the denominator. Those numbers that can't be expressed that way - for example, the square root of 2 - are said to be irrational. A rational number is any number that can be written as a ratio or fraction. If the decimal representation is finite orhas a repeating set of decimals, the number is rational. Irrational numbers cannot be reached by any finite use of the operators +,-, / and *. These numbers include square roots of non-square numbers, e.g.√2 . Irrational numbers have decimal representations that never end or repeat. Transcendental numbers are different again - they are irrational, but cannot be expressed even with square roots or other 'integer exponentiation'. They are the numbers in between the numbers between the numbers between the integers. Famous examples include e or pi (π). By definition: a rational number can be expressed as a ratio of two integers, the second of which is not zero. An irrational cannot be so expressed. One consequence is that a rational number can be expressed as a terminating or infinitely recurring decimal whereas an irrational cannot. This consequence is valid whatever INTEGER base you happen to select: decimal, binary, octal, hexadecimal or any other - although for non-decimal bases, you will have the "binary point" or "octal point" in place of the decimal point and so on. A rational number can be expressed as a fraction whereas an irrational number can't be expressed as a fraction Rational numbers can be expressed as a ratio of two integers, x/y, where y is not 0. Conventionally, y is taken to be greater than 0 but that is not an essential element of the definition. An irrational number is one for which such a pair of integers does not exist. Rational numbers can be expressed as one integer over another integer (a "ratio" of the two integers) whereas irrational numbers cannot. Also, the decimal representation of a rational number will either: terminate (eg 31/250 = 0.124); or go on forever repeating a sequence of digits at the end (eg 41/330 = 0.1242424... [the 24 repeats]); whereas an irrational number will not terminate, nor will there be a repeating sequence of digits at the end (eg π = 3.14159265.... [no sequence repeats]). Rational numbers are numbers that keeps on going non-stop, for example pie. Irrational numbers end. Its as simple as that! Improved Answer:- Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions. a rational number can be expressed as a fraction in the form a/b (ie as a fraction). a irrational number cannot be expressed as a fraction (e.g. pi, square root of 2 etc) Rational numbers can be represented as fractions. That is to say, if we can write the number as a/b where a and b are any two integers and b is not zero. If we cannot do this, then the number is irrational. For example, .5 is a rational number because we can write it as 5/10=1/2 The square root of 2 is irrational because there do not exist integers a and b such that square root of 2 equals a/b. Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions.

### Are squares of irrational numbers always rational? No. The cube root of 3, for example is irrational. But the square of cubert(3) is 3 to the power 2/3, which is irrational. Another example, pi2 is irrational (in fact so is pi to any non-zero power). 