Exponentiation

• 1. Introduction

  ›

  Exponentiation is an arithmetic operation built from just two numbers, the base and the exponent. Write a small number above and to the right of a larger one, and you have compressed an enormous idea into a single mark. When the exponent is a positive whole number, the rule is plain. You multiply the base by itself that many times. Three to the fifth power means three appears five times in the multiplication. The result is read aloud as "three to the fifth" or "three raised to the fifth power." From this modest beginning grows a structure that reaches into economics, biology, chemistry, physics, and computer science. Compound interest leans on it. So does population growth, chemical reaction kinetics, wave behavior, and the public-key cryptography that protects modern communication. The word itself comes from Latin, exponentem, the present participle of exponere, meaning "to put forth." But how did a notation so compact come to be written down? Why does anyone agree on what a number raised to a negative or fractional power should mean? And how does multiplying a number by itself end up guarding secrets across the internet? Those questions stretch from ancient Greece to the first floating-point computers, and into the strange territory where the base is a complex number with infinitely many possible answers.

• 2. A Notation Slowly Invented

  ›

  Michael Stifel coined the word exponent in 1544. The companion word "power" carries a stranger history. It descends from the ancient Greek dúnamis, meaning amplification, which Euclid used for the square of a line, following Hippocrates of Chios. The Latin terms potentia, potestas, and dignitas were a mistranslation of that Greek idea. Robert Recorde, working in the 16th century, supplied a vocabulary that now sounds like incantation. He used "square," "cube," then "zenzizenzic" for the fourth power, "sursolid" for the fifth, "zenzicube" for the sixth, "second sursolid" for the seventh, and "zenzizenzizenzic" for the eighth. The fourth power was also called "biquadrate." Notation took centuries to settle into the superscript we use today. Nicolas Chuquet used a form of exponential notation in the 15th century, and Henricus Grammateus and Stifel carried it into the 16th. Jost Bürgi, late in that century, wrote exponents as Roman numerals in a style close to Chuquet's. In 1636, James Hume used essentially modern notation in L'algèbre de Viète. Samuel Jeake introduced the term "indices" in 1696, a word once paired with "involution" before that synonym faded from use.

• 3. Counting Grains And Possessions

  ›

  In The Sand Reckoner, Archimedes proved the law of exponents needed to manipulate powers, then turned that law to a vast estimate. He used powers to count how many grains of sand could fill the universe. In the 9th century, the Persian mathematician Al-Khwarizmi gave the square a telling name. He called it māl, meaning possessions or property. Mathematicians of his era and earlier pictured a squared number as an area, especially of land, and so as property itself. For a cube he used Kaʻbah. Later Islamic mathematicians wrote these as the letters mīm and kāf, a shorthand visible by the 15th century in the work of Abu'l-Hasan ibn Ali al-Qalasadi. René Descartes set down the first form of our modern exponential notation early in the 17th century, in La Géométrie, introduced in Book I. He described multiplying a quantity by itself, then by itself again, "and thus to infinity." Descartes and others reserved exponents for powers greater than two, still preferring to write squares as plain repeated multiplication. In 1748, Leonhard Euler took a further step, introducing variable exponents and, by implication, non-integer ones. He observed that a quantity with a variable in the exponent is not an algebraic function, because in those the exponents must stay constant.

• 4. Machines And Magnitudes

  ›

  As calculation grew mechanized in the 20th century, notation bent to fit the limits of machines. The theoretical concept of floating-point representation came from the Spanish engineer Leonardo Torres Quevedo, in his 1914 Essays on Automatics. The first physical implementation arrived in 1938, when the German engineer Konrad Zuse built it into his Z1 computer. One register in Zuse's design held the leading digits, while a second held the exponent. The more flexible decimal floating-point representation followed in 1946, in a Bell Laboratories computer. Educators and engineers then adopted scientific notation, which matches how people speak of an order of magnitude on a ratio scale. The speed of light in vacuum, 299792458 metres per second, becomes a tidy 2.99792458 times a power of ten, often rounded to 2.998. In 1961, the School Mathematics Study Group developed notation tied to metric units. Exponents reached into measurement itself. Because force is mass times acceleration, it is measured in kilogram metres per second squared. Using M for mass, L for length, and T for time, dimensional analysis writes force as M L T to the negative second power.

• 5. Extending The Rules

  ›

  A nonzero number raised to the zero power equals one, a value also delivered by the empty product convention. That convention works in any algebraic structure whose multiplication has an identity. Zero raised to the zero power is the controversial case. Where only integer powers are in view, the value one is generally assigned, but elsewhere the choice depends on context. Negative exponents follow from a single identity that holds for any integer power and any nonzero base. It is the only definition that lets the multiplication rule survive into negative territory. Raising zero to a negative exponent stays undefined, though in some circumstances it is read as infinity. Fractional exponents reach back to roots. A number raised to the one-half power is the square root, the unique nonnegative real number whose square returns the original. These extensions all serve one goal, preserving the rule that powers add when you multiply. Trouble begins when the base is not a positive real number. A negative real number has a real nth root when n is odd, but none when n is even. Once complex roots enter, exponentiation with a base that is not a positive real is generally treated as a multivalued function.

• 6. Order, Growth, And Direction

  ›

  Exponentiation breaks two habits that addition and multiplication never do. It is not commutative, since swapping base and exponent changes the answer. It is not associative either, so the order of stacked powers matters. Without parentheses, serial exponentiation in superscript form is read top-down, or right-associative, not bottom-up. The behavior of power functions splits by parity. When the exponent is an even integer, the function shares the bowl shape of the square, climbing toward positive infinity in both directions and flattening more in the middle as the exponent grows. These are even functions. When the exponent is odd, the shape reverses on one side, falling toward negative infinity for decreasing values, and these are odd functions. Limits sharpen the picture. A number greater than one, raised to ever-larger powers, grows without bound. A number with absolute value less than one shrinks toward zero. Any power of one stays one, and powers of a negative number swing between positive and negative as the exponent alternates even and odd, settling on no limit. Powers of negative one drive that alternation, which makes them a clean tool for expressing alternating sequences.

• 7. Powers Of Two And Ten

  ›

  In the decimal system, a power of ten is the digit one trailed or led by zeros, with the count fixed by the exponent. SI prefixes ride on those powers. Kilo means a thousand, so a kilometre is a thousand metres. Powers of two carry their own special names at the small end, where one half and one quarter are the first negative powers. They run deep in set theory, since a set with n members has a power set of all its subsets, and that power set has two-to-the-n members. Integer powers of two anchor computer science. A positive power of two gives the number of values an n-bit binary integer can hold, so a byte takes 256 different values. The binary number system writes any number as a sum of powers of two, marked by a sequence of digits around a binary point. Tetration, the operation built from repeated exponentiation, dwarfs all of this. Evaluated at two and three, the chain of addition, multiplication, exponentiation, and tetration yields 6, 9, 27, and 7625597484987. That last leap hints at the hyperoperations expressed by the Ackermann function and Knuth's up-arrow notation.

• 8. Secrets In Finite Fields

  ›

  The Diffie, Hellman key exchange turns exponentiation into a tool for secure communication. It rests on a sharp imbalance. Exponentiation in a finite field is computationally inexpensive, while the reverse operation, the discrete logarithm, is computationally expensive. A finite field has a number of elements that is a prime or a prime power. Inside such a field, a primitive element g is one whose successive powers run through every nonzero element. If g is primitive, raising it to an exponent e can be computed efficiently by exponentiation by squaring, even when the field is large. Recovering e from the result has no known practical algorithm once the field is large enough. The same squaring trick keeps everyday computation fast. Computing two to the hundredth power by plain repeated multiplication would take 99 multiplications. Applying Horner's rule to the exponent written in binary cuts that to 8. Finding the truly minimal sequence is a hard problem, related to the subset sum problem, with no efficient algorithm known. In practice, exponentiation by squaring is fast enough and far easier to implement, which is why a difficult-looking operation quietly secures conversations that strangers can never read.