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Numbers are called transcendental. Certainly all constructible numbers are algebraic. So ? is not constructible.

Definition. A real number r ? R is called constructible if there is a finite sequence of compass-and-straightedge constructions that, when performed in order, will always create a point P with at least one coordinate equal to r.

Not all algebraic numbers are constructible. For example, the roots of a simple third degree polynomial equation x³ - 2 = 0 are not constructible. (It was proved by Gauss that to be constructible an algebraic number needs to be a root of an integer polynomial of degree which is a power of 2 and no less.)

We assume that the set of natural numbers N = { 0, 1, 2, 3, 4,... } is given. We also assume that we know all usual properties and structures. Specif- ically, N is well-ordered, it comes with addition and multiplication which satisfy usual properties like a + b = b + a for every a ? N and every b ? N.

Claim: If a and b are constructible numbers, then so is a + b. Proof: If either of a or b is zero, the claim is trivial. Otherwise, draw a line through the two points p, q that are distance |a| apart, then a circle of length |b| around q.

The real number a is said to be a constructible number if |a|, the absolute value of a, can be constructed using a compass and a straight-edge, letting 1 be the unit length. The square root of 2 is constructible since 1 is constructible.

A square with area. , the same area as a unit circle, would have side length. , a transcendental number. Therefore, this square and its side length are not constructible, because it is not algebraic over.

A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.

In general, numbers up to nine are spelled out, while those that are greater are represented by numerals. However, there are some exceptions. If you have a sentence that contains numbers that refer to the same grouping, they should appear in a consistent form.

Numbers one to nine are written as words, and 10 and more are written as digits. When writing numbers greater than 999 split the digits into groups of three separated by a comma (not a space) such as 1,999, or 25,000, or 128,282,597. The exception is street numbers and years that do not take the comma separator.