[color=#0a971e][b]Line segments are proportional to the real numbers they can represent.[/b][/color] [b]DRAG [color=#1551b5]FACTOR A •[/color] and [color=#0a971e]FACTOR B •[/color] ← or → from [color=#d69210]O[/color][/b] with your mouse. [color=#c51414][b]MOVE IMAGE[/b][/color] > Hold Shift and move the diagram on the screen with your mouse. [color=#c51414][b]ZOOM IMAGE[/b][/color] > Hold Shift and zoom in and out with your mouse wheel. [color=#c51414][b]ENTER FACTORS[/b][/color] > As well as dragging the factors left or right with your mouse you can also enter factors into the [color=#1551b5]A[/color] and [color=#0a971e]B[/color] input boxes. [b]Reset the Applet[/b] to the default ℯ × - π by clicking the icon top right [b](See further comments below the applet.)[/b]

For a fun applet showing [b]division [/b]via [b]circles[/b], goto [url]http://bit.ly/Division_with_a_Circle[/url] For a fun applet showing [b]multiplication [/b]via [b]triangles[/b], goto [url]http://bit.ly/Multiplication_with_Triangles[/url] For a fun applet showing [b]square roots [/b]via [b]semi-circles[/b], goto [url]http://bit.ly/Square_Roots_via_a_Semi-Circle[/url] [b][i]So have fun![/i][/b] Jonathan Crabtree [url]www.jonathancrabtree.com/mathematics[/url] [url]www.linkedin.com/in/jonathancrabtree[/url] P.S. Goto [url]http://bit.ly/Proportional_Co-Variation[/url] to learn more about [i][b]Proportional Co-Variation[/b][/i]. [b]Note:[/b] [i]Should A and B to be the same number, the circle cannot be created. A unique circle requires three distinct input points.[/i] ============== I attribute the idea for the above applet to Ludolph van Ceulen (1540 - 1610). A similar diagram appears in his [i]Fundamenta Arithmetica et Geometrica[/i] published 1615. This idea long predates the Cartesian Plane so all magnitudes were originally positive (signless) yet the [i]Rules of Sign[/i] may be demonstrated. Make one of the factors negative and the product on the circle is negative on the Cartesian Plane. Make both factors negative and the product is positive. It is possible van Ceulen (or someone else) got this idea by modifying proposition 35 from Book III of Euclid's Elements. Thanks to Dr David Joyce for this connection. See [url]http://bit.ly/Elements_Book_III_P35[/url] A proof of this proportional unit segment approach to multiplication by Dr Robin Hartshorne, is at [url]http://bit.ly/Multipication_via_Circle_Proof[/url] from [b]Geometry: Euclid and Beyond[/b]. [url]http://bit.ly/Geometry_Euclid_and_Beyond[/url]