Real number

• 1. Introduction

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  A real number is any number that can measure a continuous quantity, like a length, a duration, or a temperature. The word continuous hides something strange. It means that any two values can sit arbitrarily close together, with differences smaller than any gap you care to name. Stretch every such number into an infinite string of decimal digits and you have, almost uniquely, captured it. That word almost is the first clue that real numbers are subtler than they look. They are the points on a line where the integers stand equally spaced, yet between any two of them lurk numbers that no fraction can ever name. Some are roots of polynomials with rational coefficients. Others are not. For more than two thousand years, people measured, calculated, and reasoned with these numbers without ever defining what they were. How do you pin down a concept of continuity rigorous enough to build calculus on? Who finally did it, and why did it take until the second half of the 19th century? And what happens when you try to fit something infinite inside a machine that can only hold finitely many digits?

• 2. What Counts As Real

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  René Descartes, in the 17th century, reached for the adjective real to separate ordinary numbers from imaginary ones like the square roots of negative numbers. The distinction stuck. Real numbers gather the rational numbers, including the integers and fractions, into one family. Any real number that is not rational earns the name irrational. The family splits a second way. Real numbers that are roots of polynomials with rational coefficients are called algebraic numbers, and that group already contains every rational number plus many irrationals. Those reals that are roots of no such polynomial are the transcendental numbers, a stranger and more elusive group. Picture all of them laid out on the number line, also called the real line, with integer points spaced evenly along it. This was always the informal picture. The trouble is that an informal picture cannot support rigorous reasoning, and rigorous reasoning was exactly what mathematicians would soon demand.

• 3. The Rules Of Arithmetic

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  The real numbers form what mathematicians call an ordered field, and that phrase carries the whole machinery of elementary arithmetic. Two operations sit at the center: addition and multiplication. Both are commutative, so the order of two numbers does not change their sum or product. Both are associative, so the grouping of three numbers does not matter either. Multiplication distributes over addition, tying the two operations together. There is a number called zero, the additive identity, which leaves any number unchanged when added. There is a number serving as the multiplicative identity. Every real number has an additive inverse, and every nonzero real number has a multiplicative inverse. Layered on top is a total order, meaning that for any two reals exactly one of three relations holds, and the order respects both addition and multiplication. From these few rules the rest of arithmetic follows. Subtraction is just adding an additive inverse. Division is multiplying by a multiplicative inverse. The absolute value of a number measures its distance from zero, a small definition that will matter enormously once distance becomes the language of limits.

• 4. The Missing Property

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  Every property described so far is shared by the rational numbers, which means none of them distinguishes the reals at all. The line that separates the two systems is called Dedekind completeness. It states that every non-empty set of real numbers with an upper bound has a least upper bound. Take any collection of reals that does not run off to infinity, and there is always a single smallest ceiling sitting above it. The rationals fail this test. The set of rationals below the square root of 2 has plenty of rational upper bounds, such as 1.42, but no least rational upper bound, because the square root of 2 is not rational. This one property does heavy lifting. It delivers the Archimedean property, that for every real number some integer exceeds it. It guarantees that every positive real has a positive square root. It ensures that every polynomial of odd degree with real coefficients has at least one real root. Those last two facts make the reals a real closed field, which in turn yields the real form of the fundamental theorem of algebra: every real polynomial factors into pieces of degree at most two.

• 5. Infinite Decimals

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  The most familiar way to name a real number is its decimal representation, a string of digits each standing for a digit between zero and nine times a power of ten. The string runs finitely far to the left and infinitely far to the right of the decimal point. To define what such a string actually means, you truncate it after some number of places, collect all those finite truncations, and take their least upper bound. Dedekind completeness promises that this least upper bound exists. The construction can run in reverse too. Given a real number, you peel off its digits one at a time by induction, each digit the largest that keeps you from overshooting. A wrinkle appears for numbers like a terminating decimal fraction. These have two valid representations, one ending in trailing zeros and one ending in trailing nines, the situation explored under the heading 0.999. Excluding those endless-nine tails, there is an exact pairing between the real numbers and their decimal strings. None of this depends on the number ten. Swap in any base and replace the digit nine accordingly, and the same machinery holds.

• 6. Why Limits Live Here

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  A main reason to bother with real numbers is that sequences in them have limits. The formal tool is the Cauchy sequence, a notion Cauchy himself introduced. A sequence is Cauchy if its terms eventually crowd arbitrarily close to one another and stay there. In the reals, every Cauchy sequence converges, which is what it means to say the space is complete in the metric sense. The rationals lack this. The sequence 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, each term adding a digit of the square root of 2, is Cauchy yet converges to no rational number. Among the reals it converges to the square root of 2. This completeness is the bedrock of calculus and analysis. The Cauchy test lets you prove a sequence has a limit without ever computing it. The exponential function is a case in point. Its standard series converges for every input because the partial sums can be made arbitrarily small, proving the sequence is Cauchy and therefore that the function is well defined everywhere.

• 7. Counting The Uncountable

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  The set of all real numbers is uncountable. Both the natural numbers and the reals are infinite, yet no one-to-one function maps the reals into the naturals, so the reals are infinite on a larger scale. Their size is called the cardinality of the continuum. It strictly exceeds aleph-zero, the cardinality of the natural numbers, and equals the cardinality of the power set of the naturals, the collection of all their subsets. This raises a question that nearly broke set theory. Is there any size strictly between aleph-zero and the continuum? The claim that there is none is the continuum hypothesis. The standard foundation, Zermelo-Fraenkel set theory with the axiom of choice, cannot settle it. Assuming that system is consistent, the continuum hypothesis can be neither proved nor disproved within it, since some models satisfy it and others do not. Paul Cohen proved in 1963 that it stands independent of the other axioms, free to be accepted or rejected without contradiction.

• 8. A Long Road To Rigor

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  Simple fractions were already in use by the Egyptians around 1000 BC, and the Vedic Shulba Sutras of 600 BC may hold the first use of irrational numbers. Manava, who lived around 750 to 690 BC, knew that the square roots of numbers like 2 and 61 could not be determined exactly. Around 500 BC the Greeks under Pythagoras saw that the square root of 2 is irrational. For the Greeks, numbers meant only the natural numbers, and reals were treated as proportions between lengths. Eudoxus of Cnidus, around 390 to 340 BC, defined equality of irrational proportions in a way echoing Dedekind cuts more than two thousand years before Dedekind. Arabic mathematicians later merged number and magnitude, and Abu Kamil Shuja ibn Aslam, around 850 to 930, first accepted irrationals as solutions to quadratic equations. In Europe such numbers were called surd, meaning deaf. Simon Stevin built modern decimal notation in the 16th century. Lambert in 1761 gave a flawed proof that pi is not rational, Legendre completed it in 1794, Liouville established transcendental numbers in 1840, Hermite proved e transcendental in 1873, and Lindemann did the same for pi in 1882. Yet the continuity of the number line, the property now called completeness, stayed mysterious. Rigor reached numbers only in the 1800s. Cauchy made calculus rigorous in his Cours d'Analyse of 1821 while still assuming the reals undefined. In 1854 Bernhard Riemann exposed the gap through Fourier series. Then in 1872, building on work Richard Dedekind began in 1858, two definitions appeared at once, Dedekind cuts from Dedekind and equivalence classes of Cauchy sequences from Georg Cantor, the same Cantor who in 1874 showed the reals uncountable while the algebraic numbers are merely countable.