# Construction of irrational numbers

Can an irrational number be constructed which is a) not any known transcendental number b) not a surd? If yes, then how can I construct one? A detailed answer regarding the theory behind this and some references will be appreciated. I modified the a) part to what it is now, because I am guessing numbers are either algebraic or not algebraic(I.e. transcendental). Is this correct? I am thinking along the lines of constructing a sequence which converges to the desired number, but then how to construct a sequence to a desired limit?

• What is a known transcendental number BTW?
– user99914
Nov 4, 2015 at 6:15
• Err, e or pi or golden ratio? Nov 4, 2015 at 6:16
• What do you mean by a surd? If you mean something you can write down in terms of $n^{th}$ roots then the answer is yes. Some quintic polynomial should work. Nov 4, 2015 at 6:16
• @user286490 e and pi are transcendental and the golden ratio is a surd Nov 4, 2015 at 6:17
• Oh okay, and yes, by surd I mean what you. Nov 4, 2015 at 6:18

The answer is yes. If you know about Galois theory, you need an extension of $\mathbb{Q}$ that has a non-solvable Galois group (like $S_5$). If you don't know about Galois theory, then the roots of the polynomial $x^5-80x+5$ are irrational numbers but they are not surds and not transcendental.