Irrational Numbers

5.2 is rational. Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.

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Math and Arithmetic

Numbers

Irrational Numbers

If there are no numbers after the 9 it is rational

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Math and Arithmetic

Algebra

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Irrational Numbers

The length of the diagonal of any square whose sides are a whole number of units.

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Math and Arithmetic

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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.

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Math and Arithmetic

Proofs

Irrational Numbers

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.

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Math and Arithmetic

Numbers

Irrational Numbers

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.

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Algebra

Numbers

Irrational Numbers

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.

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Math and Arithmetic

Algebra

Irrational Numbers

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.)

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Irrational Numbers

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.

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Irrational Numbers

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.

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Irrational Numbers

No. If you write an irrational number as a decimal, it will have an infinite number of decimal digits that don't repeat periodically.

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Irrational Numbers

It is a real number, a rational number, an integer.

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Math and Arithmetic

Irrational Numbers

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.

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Irrational Numbers

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.

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Numbers

Irrational Numbers

4.75 is the ratio of 475 to 100 so it's rational.

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Math and Arithmetic

Numbers

Irrational Numbers

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

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Irrational Numbers

No. It can be expressed as the ratio 31/10 and is, therefore, a rational number.

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Math and Arithmetic

Numbers

Irrational Numbers

Yes. Any number that you can write down on paper with digits, and then show

it to somebody and say "There it is, that's it, that's the complete number." is a

rational number.

5.4 is the ratio of 54 to 10, and can be written as 54/10, so it satisfies the

definition of rational numbers.

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Numbers

Irrational Numbers

1.67 is the ratio of 167 to 100. That's rational.

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