What is the difference between a Surd and an Irrational number?

Question

Can you prove that square root of 3 is irrational theoratically?

Answer 1

We can prove it like this:
Suppose that the square root of 3 is r/s, where r and s are relatively prime positive integers (if the square root of 3 is rational, then we can do this).

Squaring both sides of the equation yields 3 = (r^2)/(s^2). That is,
3 s^2 = r^2.

This shows that r^2 must be divisible by 3, from which it follows that
r is divisible by 3, and r^2 is divisible by 9. This implies that s^2
must be divisible by 3, and so s must be divisible by 3. Hence, r and
s are divisible by 3. This is a contradiction to our assumption that
r and s are relatively prime. Thus, there must not exist integers r
and s so that the square root of 3 is r/s. In other words, the square root of 3 is irrational.

You can use an almost identical proof to show that the square root of any prime is irrational. With a little modification, you can use it to show that the square root of an integer is either an integer or an irrational number.

Answer 2

A surd is an irrational number of the form 2 + 3V5, etc. Surds are algebraic numbers, that is, they are the solution of a polynomial equation with integer coefficients. Not all algebraic numbers are surds, and not all irrational numbers are algebraic (e.g. pi is irrational but not algebraic).

I will prove that the square root of any prime number x is irrational.

Suppose that the square root of x is equal to a reduced fraction p/q. Then by definition

[1] ... (p/q)^2 = x
[2] ... p^2 = q^2 x

From [2] it is clear that

[3] ... p is a multiple of x

Therefore

[4] ... p^2 is a multiple of x^2

Dividing by x we find

[5] ... p^2 / x = q^2 is a multiple of x^2

Which means that

[6] ... q is a multiple of x

But [3] and [6] show that in the fraction p/q, numerator and denominator have x as a common factor, which contradicts our assumption that the fraction is reduced.

This proves that there is no fraction p/q that is equal to the square root of x.

Answer 3

i imagine it truly means a similar aspect - contained in the history of mathematics, like the Pythagoreans, "surd" meant mute. Rational numbers relate to a resonant vibration on a plucked string because the string can vibrate with N nodes the position N is the demoninator, type of. yet when the wavelength is incommensurate with the dimensions of the string then you fairly have an irrational quantity and it would not make any music. I have heard of a "surd root" and that i imagine it merely means a sq. root or dice root etc. that has an irrational fee. it will be used someplace with a extra particular meaning yet i do not comprehend a context the position that would properly be mandatory. (so some distance as my expertise of the terminology is going. you would possibly want to look it up and consider it extra.)

Answer 4

A surd is an expression whose value is irrational. A surd consists of two rational numbers.
eg. "Third(3rd) root of five(5)" is a surd.

Answer 5

surd is square root of a complex number.Irrational numbers are fractions.square root of 3 is 1.732

Answer 6

irrational numbers are any real numbers which are not in a fraction form a/b where a and b are integers.

prove that rt3 is irrational:
http://mathforum.org/library/drmath/view/52653.html

Answer 7

irrational numbers are any real numbers which are not in a fraction form a/b where a and b are integers.