1.2 Mathematics as a language 3If as many numbers as we please beginning from a unit be set out continu-
ously in double proportion, until the sum of all becomes prime, and if the sum
multiplied into the last make some number, the product will be perfect (Euclid,
Elements, IX.36, 300B.C).2nr-r ,
A - ——— = 7rrJ
The area of any circle is equal to a right-angled triangle in which one of
the sides about the right angle is equal to the radius, and the other to the
circumference, of the circle (Archimedes, Measurement of a Circle, 225B.C).S = 4wr^2
The surface of any sphere is equal four times the greatest circle in it (Archimedes,
On the Sphere and the Cylinder, 220B.C).3 In n^2 m^3 3/ n n^2 m^3Rule to solve x^3 + mx = n: Cube one-third the coefficient of x; add to it the
square of one-half the constant of the equation; and take the square root of
the whole. You will duplicate this, and to one of the two you add one-half the
number you have already squared and from the other you subtract one-half the
same... Then, subtracting the cube root of the first from the cube root of the
second, the remainder which is left is the value of x (Gerolamo Cardano, Ars
Magna, 1545).
However, the language of mathematics does not consist of formulas alone.
The definitions and terms are verbalized often acquiring a meaning different
from the customary one. In this section, the basic grammar of mathematical
language is presented.Definition 1.2.1 Definition is a statement that is agreed on by all parties
concerned. They exist because of mathematical concepts that occur repeatedly.Example 1.2.2 A prime number is a natural integer which can only be (in-
teger) divided by itself and one without any remainder.Proposition 1.2.3 A Proposition or Fact is a true statement of interest that
is being attempted to be proven.
Here are some examples:Always true Two different lines in a plane are either parallel or they intersect
at exactly one point.
Always false —1 = 0.
Sometimes true 2x — 1, by < 1, z > 0 and x,y,z e K.