Arithmetic

• 1. Introduction

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  Arithmetic is older than writing, older perhaps than language itself. The Lebombo bone, dated to about 43,000 years ago, carries notches that some historians read as the oldest arithmetic artifact known. The Ishango bone, dated between 22,000 and 30,000 years ago, tells a similar story. Both interpretations are disputed, but the impulse behind them is not. Long before anyone could write the word for a number, people were keeping track of quantities by carving marks into bone and wood.

  From those scratches grew an entire branch of mathematics. Arithmetic deals with numerical operations: addition, subtraction, multiplication, and division. In a wider sense it stretches to exponentiation, the extraction of roots, and the taking of logarithms. It is one of the earliest forms of mathematics any student ever meets, and it quietly underpins algebra, calculus, statistics, physics, and economics alike.

  How did marks on a bone become a discipline rigorous enough to need axioms? Who invented zero, and why did it arrive so late? What separates the numbers a child counts on their fingers from the numbers no fraction can ever capture? And how did the same operations that calculate change at a shop counter come to depend on a technical standard called IEEE 754? The answers run from ancient Sumeria to the inside of a microprocessor.

• 2. The Many Faces Of Number

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  The natural numbers start at 1 and run to infinity, excluding 0 and everything negative. Also called counting numbers, they answer a single question when used as cardinals: how many? Used as ordinals, the same impulse answers a different question. First, second, and third indicate order or placement in a series, telling you what position rather than what quantity.

  Whole numbers are the natural numbers with one addition: they include 0. Some mathematicians refuse to draw the line at all and simply fold 0 into the naturals. Widen the set to take in negatives, and you reach the integers, which gather positive and negative whole numbers together.

  A number is rational if it can be written as the ratio of two integers. The fraction one half is built by dividing the numerator 1 by the denominator 2. Every integer qualifies too, since each is a fraction with a denominator of 1. Decimal fractions like 0.3 and 25.12 are a special case, because their denominators are powers of 10, and every rational number corresponds to a finite or a repeating decimal.

  Irrational numbers refuse this treatment entirely. Give a right triangle two legs of length 1, and its hypotenuse becomes a number no ratio of integers can express. The number pi, the ratio of a circle's circumference to its diameter, is another. Its decimal representation runs on forever without ever repeating. The rationals and the irrationals together form the real numbers, and beyond them lie wider classes still, the complex numbers and the quaternions.

• 3. Writing Numbers Down

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  A numeral is a symbol that stands for a number, and a numeral system is the framework that governs how a handful of basic numerals combine to express any quantity. All the early systems were non-positional, meaning a digit's value never depended on where it sat. The simplest is the unary system, which uses one symbol for 1 and writes every larger number by repetition. To represent 7, you repeat the symbol for 1 seven times. Tally sticks with dents and tally marks are variations on exactly this idea.

  Egyptian hieroglyphics grew more elaborate, adding distinct symbols for 10, 100, 1000, and 10,000. The numeral for 10,405 sums one symbol for 10,000, four for 100, and five for 1. The Roman numeral system worked along similar lines, assigning I, V, X, L, C, D, and M to the values 1, 5, 10, 50, 100, 500, and 1000.

  A positional system changes everything, because a numeral's place in the expression sets its value. The system has a radix raised to a higher power at each position. In the decimal system, also called the Hindu-Arabic numeral system, the radix is 10, which is why 532 differs from 325 and 253 despite sharing the same three digits. The binary system, with a radix of 2, drives most computers, and there the number 13 is written 1101, with each digit corresponding to one bit. The earliest positional system of all belonged to the ancient Babylonians and ran on a radix of 60.

• 4. Combining And Undoing

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  Addition combines two numbers, the addends, into a single sum, and counting is simply repeated addition where 1 is added again and again. Subtraction is its inverse: take the subtrahend away from the minuend to find the difference. The two operations are so intertwined that subtraction is often treated as a special case of addition, since subtracting a positive number is the same as adding a negative one. That trick reduces the number of basic operations a calculation needs.

  Multiplication folds the multiplier and the multiplicand into a product, and when the multiplicand is a natural number it amounts to repeated addition. Division reverses it, splitting the dividend into equal parts set by the divisor to yield a quotient. Here too there is a shortcut, because dividing by a number is the same as multiplying by its reciprocal, which is 1 divided by that number.

  Exponentiation raises a base to the power of an exponent, and with a natural-number exponent it becomes repeated multiplication. Roots are exponentiation in disguise, using a fractional exponent: a square root raises a number to the power of one half. Logarithm is the inverse of exponentiation, asking which exponent a base must be raised to in order to produce a given number. Because 10 cubed is 1000, the logarithm base 10 of 1000 is 3.

  These operations obey laws. An operation is commutative if the order of its arguments can change without altering the result, as addition does. It is associative if, in a chain of two operations, it makes no difference which is carried out first, as multiplication does. Addition and multiplication each carry an identity element, 0 and 1, and an inverse element. The additive inverse of 6 is -6, since their sum is 0. Exponentiation and logarithm enjoy no such comforts: they are neither commutative nor associative, and they lack general identity and inverse elements.

• 5. Higher Arithmetic

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  Number theory studies the integers, their structure, and the laws binding them, and tradition once called it higher arithmetic. Its elementary branch tackles divisibility, factorization, and primality using elementary methods. Analytic number theory reaches instead for analysis and calculus, examining how prime numbers are distributed and the claim that every even number is the sum of two primes. Algebraic number theory brings in fields and rings, while geometric number theory studies how lattice points with integer coordinates behave in a plane. Further branches enlist probability, combinatorics, computation, and applications to physics, biology, and cryptography.

  The fundamental theorem of arithmetic anchors much of this. It states that every integer greater than 1 is either a prime number or can be written as a unique product of primes. The number 18 factors into primes, while 19 is a prime with no other factorization at all. Euclid's theorem proves that the primes never run out, that there are infinitely many of them. Fermat's Last Theorem makes a more austere claim, that no positive integer values solve a certain equation once its exponent exceeds 2.

  Multiplication of very large integers is its own frontier. Computer science hunts for algorithms with low computational complexity, and the Karatsuba algorithm, the Schonhage-Strassen algorithm, and the Toom-Cook algorithm each answer that call.

• 6. The Trouble With Precision

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  Integer arithmetic is not closed under division. Divide 7 by 2 and the answer is not a whole number but 3.5, which forces a choice. You can round, altering the original value and inviting inaccuracy, or you can divide partially and keep the remainder, calling it 3 with a remainder of 1. Rational number arithmetic sidesteps the whole dilemma, since it is closed under division as long as the divisor is not 0.

  Irrational numbers raise a harder problem, because they carry an infinite, non-repeating string of digits. The number pi begins 3.14159 and never settles. Truncating it to four decimal places keeps the leftmost digits and discards the rest, giving 3.141. Rounding to four places gives 3.142 instead, since the following digit is a 5, making 3.142 the closer approximation. These methods let computers perform fast, approximate work on real numbers.

  Science inherits a different inexactness. A height measured with a tape might be known only to the nearest centimeter, so it should read 1.62 meters, not 1.6217 meters, and should be written 1.62 plus or minus 0.005 meters when the uncertainty is made explicit. Significant digits track this honesty. The numbers 0.056 and 1200 each carry only 2 significant digits, while 40.00 carries 4. Normalized scientific notation expresses the same precision cleanly, splitting 8276000 into a significand of 8.276 and an exponent of 6.

  Computers approximate all of this with floating-point arithmetic, representing each real number through a significand, a base, and an exponent. Limited bits force rounding, and rounding breaks the rules: floating-point addition is not associative, because the errors depend on the order of the sums. The standard governing this behavior is IEEE 754, which sets how numbers are represented, how rounding is performed, and how errors and exceptions are handled.

• 7. Tools Of The Trade

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  Mental arithmetic relies on nothing but the mind, leaning on visualization, memorization, and techniques like the compensation method. To add a number near a round figure, you adjust it to the round number, do the easier sum, then correct the result afterward. The human body serves as a tool too. Finger counting introduces young children to numbers, with extended fingers standing for quantity, while the human voice does its own work in verbal counting.

  Tally marks move the record outside the body, as strokes on a surface or notches in a wooden stick, sometimes grouped in fives for easier reading. The abacus goes further, arranging beads on rods so that each moved bead counts toward a result, until the final pattern reveals the answer. Counting boards and counting rods are close relatives, the first using tokens whose value depends on their region of the board, the second using horizontal and vertical patterns.

  Sectors and slide rules refine the idea again, exploiting geometric relationships between scales to handle both basic and advanced operations, while printed tables once let people look up logarithms and trigonometric functions. Mechanical calculators automated the labor entirely, taking input through dials or keys and turning gears, levers, and wheels to compute and display. The earliest of them appeared in the 17th century, including Blaise Pascal's calculator and Gottfried Wilhelm Leibniz's stepped reckoner. Electronic calculators and computers later replaced the gears with circuits like microprocessors, transforming electric signals to do the same job.

• 8. From Tax Records To Axioms

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  Around 3000 BCE, the rise of ancient civilizations turned counting into something structured, driven by the need to track stored goods, manage land ownership, and arrange exchanges. Every major civilization built a non-positional numeral system, with symbols for addition and subtraction and an awareness of fractions, from Egypt to Sumeria, China, and India. The Babylonians took the decisive step around 1800 BCE, developing the first positional numeral system and making large numbers far easier to handle.

  The ancient Greeks shifted the entire enterprise toward abstraction. Rather than asking how numbers solved specific problems, they explored the nature of numbers themselves, validated truths with proofs, and sorted numbers into even, odd, and prime. The works of Thales of Miletus and Pythagoras in the 7th and 6th centuries BCE are often treated as the inception of Greek mathematics, and in the 3rd century CE Diophantus pushed number theory into algebraic equations.

  Zero arrived from India. Brahmagupta wrote down the exact rules for its operation around 628 CE, and he also worked through calculations with negative numbers, applying them to credit and debt, though negative numbers themselves had first been explored in Chinese mathematics in the first millennium BCE. Aryabhata treated the positional decimal system in detail around the turn of the 6th century CE. During the Islamic Golden Age, Al-Khwarizmi refined and extended that system, and his work carried the decimal numerals into a Western world still using Roman numerals, where Leonardo Fibonacci, living in the 12th and 13th centuries, helped popularize them and developed the Fibonacci sequence. In the 16th century Gerolamo Cardano conceived complex numbers to solve cubic equations.

  The foundations came last. John Napier discovered the logarithm in the 17th century. Leonhard Euler and Carl Friedrich Gauss laid the groundwork of modern number theory in the 18th and 19th centuries, while Georg Cantor's set theory and the Dedekind-Peano axioms gave natural-number arithmetic a rigorous base. Those axioms, first formulated by Richard Dedekind and refined by Giuseppe Peano, rest on just a few primitive ideas: 0, natural number, and successor. From the single rule that every natural number has a successor, and that 0 is the successor of none, the whole edifice of arithmetic can be rebuilt.