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How to Construct Number Form

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The construct state in Hebrew allows you to take two nouns and use them to create a new noun. In Hebrew, this state is called (simchat). ... The gender of a construct state noun is determined by the mismatch. For example, (“school”) is masculine because is masculine.
Of course, is not constructive because it is transcendental and so is not a root of any polynomial with rational coefficients.
Not all algebraic numbers are constructive. For example, the roots of a simple third degree polynomial equation x3 — 2 = 0 are not constructive. (It was proved by Gauss that to be constructive an algebraic number needs to be a root of an integer polynomial of degree which is a power of 2 and no less.)
All algebraic numbers are computable and therefore definable and arithmetical. The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.
The complex number i is purely algebraic. That is, we call it a “number” because it will obey all the rules we normally associate with a number. We may add it, subtract it, multiply it, and so on. The complex number i turns out to be extremely useful in mathematics and physics.
Trigonometric numbers are irrational cosines or sines of angles that are rational multiples of. Such a number is constructive if and only if the denominator of the fully reduced multiple is a power of 2 or the product of a power of 2 with the product of one or more distinct Fermat primes.
Definition 6.1. A length is constructive if it can be obtained from a finite number of applications of a compass and straightedge. A constructive number is a constructive length or the negative of a constructive length. To get things started, we assume we can construct a segment of length 1.
[Golden Ratio] The golden ratio satisfies the equation 2 1 = 0. (Refer back to Problem #6 in Exercise Set 5.5.) Show that the golden ratio is a constructive number.
The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.
In mathematics, two quantities, such as a and b, are in golden ratio if their ratio is the same as the ratio of their sum (a + b) to the largest of the quantities. Aligned in this way, a and b look pleasing to the eye because they are in golden ratio. A is approximately 1.618 times bigger than b.
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