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...
In this exploration, we will use Ford Circles to help us to better visualise these special numbers. Imagine each term of a Farey Sequence is plotted on a number line. Then draw a circle on each point with a particular radius which allows it to just touch (to kiss!) any adjacent circle.
Study the model below and use the sliders or text boxes to build your own Farey Numbers. Along the way, see if you can work out the formula for that radius, as well as watching for patterns, like the number of terms in each Farey Sequence! But be warned - not all patterns in mathematics are well-behaved!
Explore further and try building your own model for this problem using GXweb - use it to explore both the geometry and the algebra involved.
Use the MathBox to enter expressions for CAS: Define functions, simplify expressions, solve equations... For example, enter Solve(x^2-x-1=0) and press the CAS button. More?
Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.
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 "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?
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QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available.
The default link here is the Geometry Expressions Showcase, but you can use it as an alternative to sending your assessment data via email, web or Google Cloud. You can even use it to send your own messages!
Exploration 1: For example: suppose you wanted to know the next term of the sequence between two values, \(\frac{a}{b}\) and \(\frac{c}{d}\)?
Easy! Just add the numerators and the denominators! \(\frac{a+c}{b+d}\). The resulting fraction is called the mediant of \(\frac{a}{b}\) and \(\frac{c}{d}\).
Exploration 2: What do you notice when you cross-multiply numerators and denominators of pairs of Farey neighbours? (If \(\frac{a}{b}\) and \(\frac{c}{d} \)are consecutive terms of a Farey Sequence, consider \(b*c\) and \(a*d\)...)
Exploration 3: AND... If you plot the numerators against the denominators of a Farey Sequence, then you get the amazing Farey Starburst!
There are some wonderful connections between continued fractions and Farey neighbours - adjacent terms of Farey Sequences.
For example, \(\frac{2}{5}\) and \(\frac{5}{12}\) are neighbours in Farey(12), as are \( \frac{6}{19}\) and \(\frac{7}{22}\) in Farey(25). Check their continued fraction forms...
