Yesterday, during a discussion about cognition, a colleague informed me that Physicist Richard Feynman perceived colours when he saw equations. "When I see equations," he once said "I see the letters in colors - I don't know why." (Ref 1). This ability is a form of synesthesia by which there is cross-over from one sense to another. Daniel Tammet, an amazing savant, also has this gift. He "sees numbers as shapes, colors, and textures, and performs extraordinary calculations in his head", as he describes in his book, Born on a Blue Day. This suggested to me that colourised equations would be a good aid to learning for some people. (*This is just an idea. I have no evidence that it would be effective, and it could put a person who is colour blind at a disadvantage. Ref 2.*)

I took the well known roots of the quadratic equation:

and used blue to identify irrational parts and red to identify negative terms. The mapping is not clear cut, because one term is plus-or-minus, and it is only the result of the square root that is irrational.

\frac{\uc{red}{-b}\pm{\uc{blue}{\sqrt[]{(b^2 \uc{red}{- 4*a*c})}}}}{2a}

into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/. The LaTeX command

**\uc{red}{-b}**defines that the term -b is to be coloured red.

The benefits of colouring may not be apparent for a single equation. I suggest that you try it the next time you need to present a long and tedious algebraic evaluation. The low-tech approach would be to use coloured highlighting pens to mark up a printed copy of the equations. This is not a new idea, accountants have been using red ink to represent negative numbers for generations.

For the benefit anyone who is unable to view the images, the equation shown by the above images is:

(-b +/- sqrt(b^2 - 4*a*c))/2a

The negative terms -b and -4*a*c are coloured red and the sqrt() function, which may be irrational is coloured blue. The square root may also be an imaginary number, if it is the square root of a negative number.

Another application of colouring is to present each digit in its standard resistor colour code. The pattern of repeating decimals for certain fractions becomes much more memorable. For example 264/999:

This was generated from the LaTeX code:

\pagecolor{gray}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}/\uc{white}9\uc{white}9\uc{white}9 = 0.\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}...

using into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/.

Ref 1: The book Sparks of Genius, By Robert Scott Root-Bernstein. The page containing Feynman's description may be viewed in Google Books.

Ref 2: In "Colourised Equations As An Aid To Mathematics" on the Mathematics24x7 network, Bob Mathews commented that we need to create presentations that are "color-blind friendly".

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