Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

Excerpts

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  We came across equations of the form $x^{2} = -3$, to which no solutions could be assigned in terms of positive and negative real numbers.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  This is the definition of the meaning of the symbol $×$ when it is written between two ordered couples.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  Hence both for addition and for multiplication the couple $(0, 0)$ plays the part of zero in elementary arithmetic and algebra; compare the above equations with $x + 0 = x$, and $x × 0 = 0$.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  The product of the two vectors $OP$ and $OQ$ is a vector $OR$, whose length is the product of the lengths of $OP$ and $OQ$ and whose direction $OR$ is such that the angle $XOR$ is equal to the sum of the angles $XOP$ and $XOQ$.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  The answer is that it is perfectly indifferent which symbolism we adopt.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  It was receiving its final form about the same time as when the steam engine was being perfected, and will remain a great and powerful weapon for the achievement of the victory of thought over things when curious specimens of that machine repose in museums in company with the helmets and breastplates of a slightly earlier epoch.

Equations

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) × (x', y') = \{(xx' - yy'), (xy' + x'y)\}

  By definition, the product of two ordered couples is the couple whose first term is xx' - yy' and whose second term is xy' + x'y.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) × (x', y') = (x', y') × (x, y)

  The product of two ordered couples does not depend on the order of the factors.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  \{(x, y) × (x', y')\} × (u, v) = (x, y) × \{(x', y') × (u, v)\}

  Multiplying ordered couples gives the same result whichever pair of adjacent factors is multiplied first.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) × (a, b) = (c, d)

  The unknown couple (x, y) that multiplies (a, b) to give (c, d) is required to be unique (except when (a, b) is the zero couple).

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) = \frac{(c, d)}{(a, b)}

  The quotient of ordered couples (c, d) by (a, b) is the unique couple (x, y) satisfying (x, y) × (a, b) = (c, d).

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x,y) × \{(a, b) + (c, d)\} = \{(x, y) × (a, b)\} + \{(x, y) × (c, d)\}

  Multiplying an ordered couple by a sum of couples equals the sum of the separate products.

• This equation is in Imaginary Numbers (Imaginary Numbers)

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) × (0, 0) = (0, 0)

  Multiplying any ordered couple by the zero couple (0, 0) gives the zero couple.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  x × 1 = x

  Multiplying any value of x by 1 leaves x unchanged, the characteristic property of 1.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (x, y) × (1, 0) = \{(x - 0), (y + 0)\} = (x, y)

  The couple (1, 0) leaves every ordered couple unchanged under multiplication, so it is the unit couple.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  \sqrt{(-1)} × \sqrt{(-1)} = -1

  The symbol sqrt(-1) must be defined so that its square is -1; this is the property the ordered couple (0, 1) is required to have.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (0, 1) × (0, 1) = \{(0 - 1), (0 + 0)\} = (-1, 0)

  The couple (0, 1) squares to the negative unit couple (-1, 0), so it interprets the square root of -1.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (0, -1) × (0, -1) = (-1, 0)

  The couple (0, -1) also squares to (-1, 0), so it interprets the other square root of -1.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (a, 0) × (x, y) = (ax, ay)

  Multiplying a complex couple by a real couple multiplies each term by the real number a.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (0, b) × (x, y) = (-by, bx)

  Multiplying a complex couple by a pure imaginary couple (0, b) turns it through a right angle and scales it.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (a, 0) × (0, b) = (0, ab)

  Multiplying a real couple by a pure imaginary couple gives a pure imaginary couple.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (a, 0) × (a', 0) =( aa', 0)

  Multiplying two real couples gives the real couple whose term is the product of the two real numbers.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (0, b) × (0, b') = (-bb', 0)

  Multiplying two pure imaginary couples gives a real couple equal to the negative of the product bb'.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  \text{the angle } QOR = \text{the angle } XOP

  Multiplying OQ by OP rotates OQ through the angle XOP, so the angle QOR equals the angle XOP.

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  (u, v) + (3, 0) = (2, 0)

  The equation x + 3 = 2 becomes this equation for the unknown couple (u, v), where x is represented by (u, v).

• Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})

  \{(u, v) + (3, 0)\}^{2} = (-2, 0)

  The equation (x + 3)^2 = -2 becomes this equation for the unknown couple (u, v).

Problems

No exercises in this chapter.