The Symbolism of Mathematics

Excerpts

• The Symbolism of Mathematics

  By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.

• The Symbolism of Mathematics

  Operations of thought are like cavalry charges in a battle---they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

• The Symbolism of Mathematics

  This service is performed by $0$, the symbol for zero.

• The Symbolism of Mathematics

  Similarly the important way of writing the equation $x = 1$ is $x - 1 = 0$, and of representing the equation $3x - 2 = 2x^{2}$ is $2x^{2} - 3x + 2 = 0$.

• The Symbolism of Mathematics

  Mathematicians have chosen to make their symbolism more concise by defining $xy$ to stand for $x × y$.

• The Symbolism of Mathematics

  Thus when we substitute $2$ for $x$ and $3$ for $y$ in $xy$, we must write $2 × 3$ for $xy$, and not $23$ which means $20 + 3$.

• The Symbolism of Mathematics

  Thus, by now varying $a$, $b$, and $c$, we arrive at the idea that $ax + by - c = 0$ represents a variable linear correlation between $x$ and $y$.

Equations

• This equation is in Variables (Variables)

• The Symbolism of Mathematics

  (x + y) + z = x + (y + z)

  When adding three numbers, the grouping of the two additions does not change the sum.

• The Symbolism of Mathematics

  x × y = y × x

  Multiplying x by y gives the same result as multiplying y by x.

• The Symbolism of Mathematics

  (x × y) × z = x × (y × z)

  When multiplying three numbers, the grouping of the two multiplications does not change the product.

• The Symbolism of Mathematics

  x × (y + z) = (x × y) + (x × z)

  Multiplying a number by a sum equals the sum of the two products with that number.

• The Symbolism of Mathematics

  x + y - 1 = 0

  The standard form of the correlation x + y = 1, with everything moved to the left so the right side is zero.

• The Symbolism of Mathematics

  ax + by - c = 0

  The general linear relation between variables x and y, with a, b, c as parameters (constants for a given relation) that can be varied to give any linear correlation.

• The Symbolism of Mathematics

  0 × x = 0

  Multiplying any number by zero gives zero.

• The Symbolism of Mathematics

  x + 0 = x

  Adding zero to any number leaves the number unchanged.

• The Symbolism of Mathematics

  x^{2} + (0 × x) - 4 = 0

  The equation x^2 - 4 = 0 written with the zero term 0 × x inserted, so that it has the same algebraic form as the other quadratic equations.

Problems

No exercises in this chapter.