Generalizations of Number

Excerpts

• Generalizations of Number

  The Greeks thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of $2$ to $3$.

• Generalizations of Number

  For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is $\sqrt{2}$ times the length of the side. But there is no fraction which exactly represents $\sqrt{2}$.

• Generalizations of Number

  One very simple way of doing this is to add the fractions together and to halve the result.

• Generalizations of Number

  But if we now interpret our symbols as “operations,” all limitation vanishes like magic.

• Generalizations of Number

  If a balance at the bank is positive, an overdraft is negative.

• Generalizations of Number

  Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct.

Equations

• Generalizations of Number

  x + a = b

  The general equation of the form x + a = b, with a and b any constants, which the book uses to show why operations remove the need for limits on the constants.

• Generalizations of Number

  x = b - a

  The solution of the general equation x + a = b is x = b - a, an operation of addition or subtraction as the case may be.

Problems

No exercises in this chapter.