# Multiplication using Parabola

## Theorem

Let the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane.

Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be points on the curve $\map f x$ so that $x_a < x_b$.

Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$.

Thus $P$ can be used as a nomogram to calculate the product of two numbers $x_a$ and $x_b$, as follows:

$(1) \quad$ Find the points $-x_a$ and $x_b$ on the $x$-axis.
$(2) \quad$ Find the points $A$ and $B$ where the lines $x = -x_a$ and $x = x_b$ cut $P$.
$(3) \quad$ Lay a straightedge on the straight line joining $A$ and $B$ and locate its $y$-intercept $c$.

Then $x_a x_b$ can be read off from the $y$-axis as the position of $c$.

## Proof

Let $\map f x = x^2$.

Then:

$\map f {x_a} = x_a^2$

and:

$\map f {x_b} = x_b^2$

Then the slope $m$ of the line segment joining $A B$ will be:

 $\ds m$ $=$ $\ds \frac {x_b^2 - x_a^2} {x_b - x_a}$ Equation of Straight Line in Plane: Point-Slope Form $\ds$ $=$ $\ds \frac {\paren {x_b - x_a} \paren {x_b + x_a} } {x_b - x_a}$ Difference of Two Squares $\ds$ $=$ $\ds x_b + x_a$ cancelling, $x_a \neq x_b$
$y = \paren {x_b + x_a} x + c$

where $c$ denotes the $y$-intercept.

Substituting the coordinates of point $A = \tuple {x_a, x_a^2}$ for $\tuple {x, y}$:

 $\ds x_a^2$ $=$ $\ds \paren {x_b + x_a} x_a + c$ $\ds \leadsto \ \$ $\ds c$ $=$ $\ds x_a^2 - \paren {x_a + x_b} x_a$ $\ds$ $=$ $\ds x_a^2 - x_a^2 - x_b x_a$ $\ds$ $=$ $\ds -x_b x_a$

$\blacksquare$