August 26 – Happy Birthday, Johann Heinrich Lambert

Postednon August 26, 2014

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nYounknow how we love celebrating Pi Day every March 14? Well, today wencan eat pie again—this time, birthday pie in honor of the fellownwho proved that pi is an irrational number!

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JohannnHeinrich Lambert, born on this date in 1728, was a Swissnmathematician, physicist, philosopher, and astronomer. n

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nLambertndid a lot of cool stuff in math, including working with non-Euclideanngeometry…that is, the kind of geometry that deals with curved space. He also studied conic sections and helped make thencalculation of the orbits of comets simpler.

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nLambertnalso studied map projections and showed that map makers could not getnBOTH the outlines of landforms AND the size (or area) of thosenlandforms right, because the Earth is round (almost spherical) andnmaps are flat (pretty much two-dimensional).

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nInnphysics, Lambert studied light and perspective and optics and color.nIn astronomy, Lambert developed theories about the generation of thenuniverse and about star systems. He wrote about logic and philosophy,nand he worked with famous philosopher Immanuel Kant.

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nButnpi…ah, pi! n

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nLet’sntalk about pi!

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nWhatndoes it mean to say that pi is an irrational number?

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nRationalnnumbers are those that can be expressed as a fraction. The numbern124 is rational because it can be expressed as a fraction:

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n124

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nOne-half is rational because it can be expressed as a fraction:

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Butnpi cannot be expressed as a fraction. You may remember that pi is thenanswer to the problem of dividing a circle’s circumference by itsndiameter. EVERY SINGLE CIRCLE – no matter what it’s circumferencenand diameter – when you divide the former by the latter, you comenup with the same exact number….a number that cannot be expressed asna fraction.

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n(Pinis close to 22/7 – but close is not the same as equal, in math.)

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nAnothernway of talking about rational and irrational numbers is to explain that a rational number cannbe expressed as a decimal, such as these decimal numbers:

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n6.78

n0.34

n8n(which is the same as 8.0)

n0.125635895322

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nSomenrational numbers are not as simple as these, and they can benrepresented by decimal numbers that NEVER END but instead go on andnon and on and on forever.

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nHerenis one:

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nThisnrational number, one-third, can be represented by this decimalnnumber:

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n0.33333333333333333…

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nEXCEPTnto make it accurate, I would have to keep typing 3s forever, and younwould have to keep reading 3s forever, and neither of us would getnanything else done. Since that would be boring, we call thesendecimals that go on forever “repeating decimals.” And we writenrepeating decimals by either:

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n puttingn three dots after the part that repeats,

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n orn putting a little line over the top of the part of the decimal thatn repeats forever.

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nHerenare some more repeating decimals:

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nNow…that’snall rational numbers.

nWhatnabout irrational numbers?

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nWhennyou try to show an irrational number as a decimal number, the numbersngo on and on forever BUT DON’T REPEAT!

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nHerenis a little bit of pi:

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nMathematiciansnhave used computers to figure out the digits of pie out to more thann10 trillion digits! And there is no repeating pattern. We can say that pi isnsorta kinda close to 3.14 – but remember, in math “close” isnnot the same as “equal.”

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nToday’snbirthday boy, Lambert, is the first mathematician to offer anmathematical proof that, no matter how far into pi you go, there willnnever be repeating decimals. In other words, he proved that thennumber is irrational.

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nAlsonon this date: