40 Fun Facts About Exponents

Which name did a nine-year-old boy coin for the number 10¹⁰⁰?

Googol was the invention of Milton Sirotta, nephew of mathematician Edward Kasner, and the search engine Google later took its name from a respelling of it.

In a famous legend about doubling, grains of wheat are laid one, two, four and so on across the squares of what?

A chessboard filled this way would hold 2⁶⁴ − 1 grains, more than 18 quintillion, and in some tellings the ruler has the inventor who asked for them executed.

Which French philosopher, namesake of Cartesian coordinates, introduced today's raised-number notation for powers?

René Descartes set out the notation in La Géométrie in 1637, yet he kept writing squares the long way, as a letter repeated twice.

Any nonzero number raised to the power of zero, whether 9⁰ or 25⁰, always equals which value?

1 is the result because dividing a power by itself, such as 9² ÷ 9², subtracts the exponents to leave 9⁰, and any number divided by itself is 1.

Which rule of thumb, named after an Intel co-founder, says transistor counts on a chip double about every two years?

Moore's law began as a 1965 magazine forecast of doubling every year, an exponential pace that Gordon Moore himself slowed to every two years in 1975.

In Einstein's famous equation E = mc², the exponent 2 tells you to square which physical quantity?

Speed of light squared is so enormous that a single kilogram of mass corresponds to roughly 10¹⁷ joules of energy.

Which scale, introduced in 1935, rises by one whole number for every tenfold jump in recorded wave amplitude?

Richter magnitudes hide even bigger leaps in energy, since each whole-number step means about 31.6 times more energy released.

Which household product did Britney Gallivan fold in half twelve times in 2002, beating a supposed limit of eight?

Toilet paper did the job, a single strip 4,000 feet long, because every fold doubles the layers and twelve folds stack 4,096 of them.

Which French lawyer wrote in a book margin that aⁿ + bⁿ = cⁿ has no whole-number solutions for exponents above 2?

Pierre de Fermat left no proof behind, and the claim resisted every mathematician for 358 years until Andrew Wiles settled it in 1994.

Which puzzle of discs stacked on three pegs takes at least 2ⁿ − 1 moves to solve when it has n discs?

Tower of Hanoi went on sale in 1883 with a legend of temple priests shifting 64 golden discs, a task whose completion would end the world.

Multiplying powers of one base, three a's then four more, a³ × a⁴ simplifies to which power?

a⁷ is right because the bases match, so the exponents add: three factors of a and four more make seven in all. Multiplying 3 by 4 to get a¹² is the classic slip.

Raising a power to another power, as in (y²)⁵, five groups of two y's, gives which simplified expression?

y¹⁰ follows because raising a power to a power multiplies the exponents, 2 × 5. Adding them to get 7 confuses this with the rule for multiplying like bases.

Dividing powers of one base, m⁸ ÷ m² cancels two m's from the top and leaves which power?

m⁶ is what remains, since dividing like bases subtracts the exponents, 8 − 2. Dividing the exponents instead gives the tempting wrong answer m⁴.

In the power 5⁻², which equals 1/25, the negative exponent tells you to take which of these?

The reciprocal is what a negative exponent calls for, so 5⁻² flips to 1 over 5². The result here is a small positive fraction, not a number below zero.

Written as a decimal, 10⁻³ equals which number, the same size as the metric prefix milli?

0.001 is one thousandth, because 10⁻³ means 1 divided by 10³, which is why a millimetre is a thousandth of a metre.

A negative base raised to an even exponent, such as (−2)⁴, always gives which kind of result?

Positive is guaranteed, because the minus signs cancel in pairs: (−2)⁴ is 16. That is also why no real number squared can come out below zero.

In written mathematics, −3² with no brackets is read as which value, since the exponent acts before the minus sign?

−9 is the standard reading, yet type −3^2 into Microsoft Excel and the spreadsheet squares the negative number first and returns 9.

The exponent in (3x)² applies to every factor inside the brackets, so the expression expands to which of these?

9x² is the expansion, since both the 3 and the x are squared. Squaring only the x gives 3x², the slip the brackets are there to prevent.

The mnemonic PEMDAS, whose E stands for exponents, helps students remember which set of rules?

Order of operations puts exponents second, straight after parentheses, and British schools teach the same sequence as BIDMAS, with I for indices.

Which way of writing numbers turns 300,000,000 into 3 × 10⁸, using a power of 10 to keep huge values short?

Scientific notation goes by the name standard form in Britain, and calculators shorten it with the letter E, so 3 × 10⁸ can show as 3E8.

A byte holds eight bits, each a 0 or a 1, so 2⁸ gives how many different values it can store?

256 values run from 0 to 255, and the eight-bit byte became the standard partly because eight is itself a convenient power of two.

IPv4 internet addresses are 32 bits long, so 2³² allows roughly how many unique addresses?

4.3 billion is fewer than one address for every person on Earth, which is why the pool ran dry and a successor with 128-bit addresses, IPv6, was drawn up in the 1990s.

One bacterium splits in two every 20 minutes; after three hours of such doubling, how many bacteria are there?

512 bacteria is 2⁹, one doubling for each of the nine 20-minute periods in three hours. Three more hours at that pace would take the count past a quarter of a million.

After three half-lives, a radioactive sample has halved three times over; which fraction of the original is left?

One eighth is left, because each half-life multiplies what remains by ½ and (½)³ is 1/8. Wait ten half-lives and less than a thousandth survives.

Dating ancient bones and wood relies on the exponential decay of which isotope, which halves about every 5,730 years?

Carbon-14 dating won Willard Libby the 1960 Nobel Prize in Chemistry, and it stops working at about 50,000 years, when too little of the isotope is left to measure.

Under the inverse-square law, moving three times farther from a lamp cuts its brightness to which fraction?

One ninth is all that arrives, because the same light spreads over a sphere whose area grows with the square of the distance, and 3² is 9.

Which finance shortcut divides a number by the interest rate to estimate the years money needs to double?

Rule of 72 works because compound interest is exponential growth: at 6 percent a year, savings double in about 12 years.

Doubling every edge of a cube multiplies its volume by how much, because 2 is raised to the third power?

8 times the volume comes with only 4 times the surface area, a mismatch that Galileo first described in 1638 and that still limits how far aircraft can be scaled up.

Which design couple, makers of a famous lounge chair, directed the 1977 film Powers of Ten about cosmic scale?

Charles and Ray Eames pull the camera back ten times farther every ten seconds from a Chicago picnic, then dive inward to the quarks inside a single proton.

The number 6.022 × 10²³, the count of particles in one mole of a substance, honours which Italian scientist?

Amedeo Avogadro never saw the number that carries his name: he died in 1856, nine years before anyone first estimated it.

Which Swiss mathematician's identity, e^(iπ) + 1 = 0, was voted the most beautiful theorem in mathematics?

Leonhard Euler's identity ties together five fundamental constants, yet he may never have written it down in this compact form himself.

Which English scientist extended the binomial theorem to fractional and negative exponents?

Isaac Newton found the general version in 1664–65, while still in his early twenties, building on John Wallis's method of interpolation.

Which Indian mathematician told G. H. Hardy that taxi number 1729 was the smallest sum of two cubes in two ways?

Srinivasa Ramanujan was ill in hospital when Hardy called the number dull; 1729 is 1³ + 12³ and also 9³ + 10³.

Which ancient Greek wrote The Sand Reckoner, inventing names for huge powers to count grains filling the universe?

Archimedes sized his universe with the sun-centred model of Aristarchus, in a work only about eight pages long addressed to a king of Syracuse.

The tempting but false rule (x + y)ⁿ = xⁿ + yⁿ is known to mathematicians by which nickname?

Freshman's dream has a true counterpart named the sophomore's dream, a pair of genuine identities that Johann Bernoulli found in 1697.

Primes that are one less than a power of two, such as 31 and 127, are named after which French friar?

Marin Mersenne studied them in the early 1600s, and because numbers of his form are easier to test, many of the largest known primes belong to the family.

Which operation, named by Reuben Goodstein from the Greek for four, means repeated exponentiation?

Tetration grows explosively: 3 tetrated to 3 is a tower of three 3s, which works out to 7,625,597,484,987.

Robert Recorde's 1557 word zenzizenzizenzic, built from an old term for squared, named which power of a number?

The 8th power is a square of a square of a square, and Recorde's coinage survives as an oddity with more Zs than any other word in the Oxford English Dictionary.

Which computer scientist, author of The Art of Computer Programming, devised up-arrow notation for towers of exponents?

Donald Knuth introduced the arrows in 1976 for numbers too large to write with ordinary exponents, and he also created the TeX typesetting system.

Which German monk, who also predicted the world would end in 1533, coined the mathematical word exponent?

Michael Stifel introduced the word in 1544 in his book Arithmetica integra, and after his doomsday forecast failed he never made another prediction.