AlgebraExplorer

Use the MathQuill MathBox to quickly and easily enter, transform and explore numbers and algebraic terms. You can even add multiple steps using the MathBox and check that each step follows from those before.

Hint: When entering mathematical expressions in the MathBox, use the space key to step out of fractions, powers, etc.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot" with a backslash (\)): for example: "\sqrt(2)[space][space]" or "\nthroot [space] n [space] x [space]".

More worked examples using Mathquill...

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Explore Continued Fractions

Try wrapping cf(...) around your fractions, decimals, irrationals - even sums and integrals. This is a great way to calculate both exact values and approximations of rationals and irrationals - but, be warned - you may never look at numbers quite the same way again...

For example, try the Continued Fraction of the Integral of the nth root of x from 0 to m \(ContinuedFraction\left(\int_0^{m}\left(\sqrt[n]{x}\right)dx\right) \): visualise the continued fraction, and then press Input ⇒ CAS . Try different values for m and n.

Tap the image to see an example.

Continue to play here, or visit the Continued Fraction Explorer page.

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Farey Sequences and Ford Circles

Suppose you wanted to make a list of all the rational numbers between 0 and 1. How would you start?

One approach might be to begin with the denominators of the fractions: first, list those with denominator 1 - zero and 1: \(\frac{0}{1}, \frac{1}{1} \).

Next, add those with denominator 2: \(\frac{0}{1}, \frac{1}{2}, \frac{1}{1} \).

Add those with denominator 3 ( \(\frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1} \)) and you are on your way to the Farey Sequence! This last would be referred to as the Farey Sequence of Order 3.

So many patterns! So much to explore...

For example: try Farey(n) - and then try different values for n!

Tap the image to see examples.

Continue to play here, or visit the Farey Sequences and Ford Circles Explorer page.

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GXWeb: Geometric Algebra or Algebraic Geometry?

For a whole different approach to algebra, explore GXWeb, an amazing blend of constraint-based dynamic geometry and computer algebra. Build your algebraic forms geometrically (or just sketch a graph!), copy the code for the expression or equation, and paste it into the MathBox below - then transform and explore using the tools available here.

Tap the image to see a geometric example.

More Mathquill examples...

Algebra Explorer

Getting Started

Explore Continued Fractions

Farey Sequences and Ford Circles

GXWeb: Geometric Algebra or Algebraic Geometry?

Some Worked Examples to Try...

About this Utility