One of the oldest problems in mathematics is squaring the circle: constructing a square with the same area as a circle. Another famous problem is doubling the cube. In each case, the aim is to construct a given number with straightedge and compass: π or .
This naturally raises a question: which numbers can be "constructed with straightedge and compass"? The answer did not emerge until the 19th century.
A little history
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What is a number? In antiquity, number and measurement were treated as one and the same. This created difficulties: how could the "product" of two numbers be defined? Representing the product of two numbers (measurements) gives an area; the product of three gives a volume; and multiplying two areas gives... something rather strange! The quotient of two numbers, meanwhile, may be a measurement—the quotient of an area by a length—or a dimensionless number if both numbers are lengths.
It was not until René Descartes published his famous Géométrie in 1637 that numbers entered geometry, through an idea as simple as it was brilliant: the French mathematician and philosopher chose a line segment, assigned it length 1, and then expressed measurements—lengths, areas, volumes and so on—relative to this standard unit. Thus, to "construct" a number is to construct a line segment whose length is that number. Then, almost surreptitiously, Descartes introduced what we now call Cartesian coordinates, allowing a point in the plane to be "represented algebraically" by two numbers.
