About this book
Contents
Section 1: Fractions old-style
When talking about fractions, we often begin by evoking the idea of fair sharing. Yet this approach does not do justice to the richness of the uses we can make of them.
Topics covered
The case of the camels Geopolitics of fractions in China The rule of three, the most beautiful of all rules? Proportionality: so crucial, so difficult The Descartes way Egyptian style Facing decimal expansions Fractions and music: when 3/4 is not equivalent to 6/8 Section 2: Selected pieces
While the importance of fractions such as 1/2 or 365 + 1/4 is easy to grasp, one is less prepared to see -1/12 emerge at the heart of number theory, or to discover so many wonderful properties associated with the decimal expansion of 1/7.
Topics covered
256/81, the π number of ancient Egypt What 1/7 has to say in base ten 365 + 1/4 and then some The astonishing presence of -1/12 The quest for 1/137 Fractions in the measurement of the circle
Section 3: The reign of the periodic
It is only if the return times of planets to their initial positions are in rational ratios that the cosmos will return to its initial state. From the hunt for exoplanets to billiards, the kingdom of fractions is that of the eternal return.
Topics covered
The Great year against incommensurability Plane tilings and fractions The ballet of the planets Planetary resonances Rational billiards Gears
Section 4: Fractions to observe
The value 1/3, whose decimal expression has no end (0.333...), offers a simple example of the superiority of fractional notation over decimal notation. The decimal expansion has not had its last word, however.
Topics covered
Molecules reveal their ratios The failure of fractions Heron: the irrational seen through fractions To infinity without rushing Fractions have team spirit Fractional analysis
Section 5: When fractions continue
Fundamentally confined to the set of rational numbers, fractions nonetheless manage to escape it when we let them continue to infinity. There are multiple ways to do this
Topics covered
Sums without end Infinite denominators Egyptian series Engel series Fractions of fractions without restraint Genealogy of fractions
Section 1: Fractions old-style
When talking about fractions, we often begin by evoking the idea of fair sharing. Yet this approach does not do justice to the richness of the uses we can make of them.
Topics covered
The case of the camels Geopolitics of fractions in China The rule of three, the most beautiful of all rules? Proportionality: so crucial, so difficult The Descartes way Egyptian style Facing decimal expansions Fractions and music: when 3/4 is not equivalent to 6/8 Section 2: Selected pieces
While the importance of fractions such as 1/2 or 365 + 1/4 is easy to grasp, one is less prepared to see -1/12 emerge at the heart of number theory, or to discover so many wonderful properties associated with the decimal expansion of 1/7.
Topics covered
256/81, the π number of ancient Egypt What 1/7 has to say in base ten 365 + 1/4 and then some The astonishing presence of -1/12 The quest for 1/137 Fractions in the measurement of the circle
Section 3: The reign of the periodic
It is only if the return times of planets to their initial positions are in rational ratios that the cosmos will return to its initial state. From the hunt for exoplanets to billiards, the kingdom of fractions is that of the eternal return.
Topics covered
The Great year against incommensurability Plane tilings and fractions The ballet of the planets Planetary resonances Rational billiards Gears
Section 4: Fractions to observe
The value 1/3, whose decimal expression has no end (0.333...), offers a simple example of the superiority of fractional notation over decimal notation. The decimal expansion has not had its last word, however.
Topics covered
Molecules reveal their ratios The failure of fractions Heron: the irrational seen through fractions To infinity without rushing Fractions have team spirit Fractional analysis
Section 5: When fractions continue
Fundamentally confined to the set of rational numbers, fractions nonetheless manage to escape it when we let them continue to infinity. There are multiple ways to do this
Topics covered
Sums without end Infinite denominators Egyptian series Engel series Fractions of fractions without restraint Genealogy of fractions