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Which Numbers Can You Construct?

Lay a number line under your drawings and the compass and straightedge stop being toys — they become a calculator. We find exactly which lengths they can produce: start from 1, add, subtract, multiply, divide, and take square roots, and you have the whole story.

From Pictures to Numbers

In the first three guides of this rung you learned to do things — bisect, copy, drop a perpendicular, raise a parallel, inscribe a hexagon. Each move turns given points into new ones with nothing but a compass and an unmarked edge. Now we ask a different kind of question, the question that turns a craft into a theory: of all the lengths in the world, exactly which ones can a finished construction hand you? To answer it we stop thinking in pictures and start thinking in numbers.

The trick is small and powerful: drop a number line under your figure. Mark a point and call it 0, mark a second point and declare the segment between them to have length 1 — this is your unit, the one length you are given for free. From here on, a length is a number, and a number is a length. A segment of length 3 is just three copies of the unit laid end to end, which you already know how to make with the copy-a-segment construction. The question "which numbers can I construct?" now has teeth.

A length is, of course, never negative, but it is convenient to call a number constructible if you can construct a segment of that absolute length. So when we say -2 or 1/3 is constructible we mean: a segment of length 2, or of length 1/3, can be built from the unit. The sign and the direction are bookkeeping you add by choosing which way to lay the segment on the line.

The Five Things a Straightedge and Compass Can Do to Numbers

Here is the heart of it. Starting from your unit length 1, the constructions you already own let you carry out exactly five arithmetic operations on lengths, and no others: addition, subtraction, multiplication, division, and the taking of a square root. The first two are obvious — to add lengths a and b, copy b onto the end of a; to subtract, copy it back the other way. The surprises are the last three, because multiplying and dividing two lengths, or square-rooting one, sound like they should need a ruler with numbers on it. They do not. They need similar triangles.

Multiplication and division both ride on one idea you proved a whole rung ago: in similar triangles, corresponding sides are in proportion. Set up two lines crossing at a point O. On one line mark the unit length 1 and, further out, the length a. On the other line mark the length b. Now draw the segment joining the tip of the unit to the tip of b, and through the tip of a draw a parallel to it. Where that parallel meets the second line, it cuts off a length x. Because the two triangles are similar, 1 : a = b : x, which rearranges to x = a times b. You have multiplied two lengths with nothing but parallels.

Multiply:   1 : a  =  b : x   ->   x = a*b
Divide:     a : 1  =  b : x   ->   x = b/a
Square root (geometric mean, altitude on a diameter):
            laying 1 and a in a row, altitude h obeys
            h^2 = 1 * a   ->   h = sqrt(a)
The three non-obvious operations, each one a single proportion read off a pair of similar triangles.

Division is the same picture run with the roles swapped, giving x = b / a. The square root is the prettiest. Lay the unit segment of length 1 next to a segment of length a in a straight row, so the whole stretch has length 1 + a, and draw the circle with that stretch as diameter. At the point where 1 meets a, erect a perpendicular up to the circle; call its height h. By the right-triangle relations on a semicircle — the same Pythagorean family you know — that altitude is the geometric mean of the two pieces, so h^2 = 1 times a, that is h = sqrt(a). A square root, built from a half-circle and one perpendicular.

The Constructible Numbers, Named

Now collect everything you can reach. Begin with 1. By adding it to itself you reach every whole number; by dividing you reach every fraction p/q. So every rational number is constructible — that is the floor you stand on for free. The five operations then let you climb: from rationals you can take a square root and land on something new, like sqrt(2) or sqrt(5), then add, multiply, and divide those with rationals and with each other, then square-root again. A constructible number is precisely any number you can write starting from rationals using only +, -, *, /, and sqrt, nested as deeply as you like.

A monster like sqrt(1 + sqrt(2 + sqrt(3))) is constructible: each layer is one of the five legal moves, so you build it from the inside out, one similar-triangle or one semicircle at a time. This is why constructions can feel almost limitless — the tower of nested roots never has to stop. The deep fact, which we will not prove here because the honest proof belongs to a later rung in coordinate and algebraic methods, is the converse: these towers of square roots are all you can ever reach. Nothing a compass and straightedge produce escapes the family +, -, *, /, sqrt.

A Number You Can Build, Step by Step

Abstraction is cheap; let us actually construct a length. Take the golden ratio phi = (1 + sqrt(5)) / 2, the number that rules the regular pentagon. Read it as a recipe in the five operations: take 5, square-root it, add 1, divide by 2. Every one of those is a move you now own, so phi must be constructible — and indeed the regular pentagon, which secretly contains phi in its diagonals, is one of the classical constructible figures. Watching the recipe become a drawing is the whole point of this guide.

  1. Lay out the unit. Mark 0 and 1 on a line; this fixes the length you call 1.
  2. Build the length 5 by copying the unit segment end to end five times with copy-a-segment (or build 2 and 1 and combine — any rational is free).
  3. Take the square root: place 1 next to 5, draw the semicircle on the combined diameter of length 6, and erect the perpendicular at the join. Its height is sqrt(5).
  4. Add 1: copy the unit onto the end of the sqrt(5) segment to get 1 + sqrt(5).
  5. Divide by 2 by bisecting that segment once — the equal-division construction with n = 2. The half-length you read off is exactly phi.

Notice that nothing in those five steps used a number printed on a tool. Every length came from copying, a semicircle, or a bisection — pure compass and straightedge. That is what it means for a number to be constructible: not that you can compute it, but that you can lay down a segment whose length equals it, exactly, with the two ancient instruments. The recipe in symbols and the drawing on paper are two faces of one object.

What This Quietly Rules Out

A theory that says exactly what you can build also says, by its silence, what you cannot. If every constructible number lives in a tower of square roots, then any length that provably needs a cube root — or worse, no roots at all — is forever out of reach with these tools. This is the hinge on which the next guide swings. Doubling a cube asks for a side scaled by the cube root of 2, an honest cube root that no stack of square roots can equal; that is why doubling the cube is impossible by compass and straightedge.

Squaring the circle is ruled out even more decisively. A square with the same area as a unit circle would need a side of length sqrt(pi), so you would first have to construct pi itself. But pi is transcendental — it satisfies no polynomial equation with whole-number coefficients at all, never mind a tower of square roots — so it lies entirely outside the constructible family. That is the deep reason squaring the circle cannot be done. The proof that pi is transcendental is real machinery, well beyond this rung, and we will be honest about that rather than wave it away.