Intellect PlusBasics
347 cards. Each one is a 40-second story with a five-minute read behind it. Page 6 of 6.
Pi's transcendence forbids squaring the circle by compass alonePi (π) is the constant ratio of a circle's circumference to its diameter—about 3.14159… Irational and transcendental, it never repeats as a decimal and is not a root of any nonzer…
Place value lets the same digit mean ones or hundredsPositional notation makes a symbol's value depend on place as well as shape—each slot a power of the base. Roman numerals mostly add fixed symbols instead. Babylonians pioneered b…
Rationals are exactly the fractions of two integersA rational number is a quotient p/q of integers with q nonzero—every integer included when q is 1. The set of rationals closes under addition, subtraction, multiplication, and div…
Reals measure continua that rationals alone cannot fillReal numbers measure continuous one-dimensional quantities—length, time, temperature—where values can differ by arbitrarily small amounts. Each has an essentially unique infinite …
Rome counted with additive letters until place value won outRoman numerals dominated European number writing into the Late Middle Ages, summing symbols like C for 100. From the 1400s, Hindu–Arabic place value gradually replaced them, yet y…- π is fixed by a circle; some constants are just famousA mathematical constant is a number locked by a clear definition and often nicknamed with a letter or a mathematician’s surname. Some, like π, fall out of geometry itself. Others …
- Mathematics proves by reason what science checks by measurementMath studies abstract objects—numbers, shapes, sets, functions, probabilities—using logic and proof rather than experiment. It models the world constantly, yet its theorems stand …
- Clock math is modular arithmetic in plain sightModular arithmetic lets integers wrap around a chosen modulus, the way a 12-hour clock turns 7 plus 8 into 3 instead of 15. Carl Friedrich Gauss systematized the modern number-the…
- Multiplication began as disciplined repeated additionAmong arithmetic’s four elementary operations, multiplication yields a product, written with a cross, a dot, juxtaposition, or a programming asterisk. For whole numbers it means a…
Natural numbers can’t even agree whether zero countsNatural numbers are the counting sequence—starting at 0 or at 1, depending who you ask—denoted ℕ in bold or blackboard type. They label how many (cardinals) and which place in lin…
Diophantus called negative answers absurd, yet Han China already used themA negative number sits below zero—the opposite of a positive quantity—handy for debts, losses, or opposite charges. Han-era Nine Chapters on the Mathematical Art already used them…
An nth root undoes raising a number to the nth powerIf n copies of r multiply to x, then r is an nth root of x. Degree two is a square root; degree three a cube root; higher degrees take ordinal names. Extracting roots is the inver…- Zero forced philosophers to treat nothing as a valueNumbers let us count, measure, order, and label—from natural 1, 2, 3 onward to zero, negatives, rationals, reals like √2 and π, and complexes built with a square root of −1. Accep…
The symbols “11” mean eleven, three, or two by baseA numeral system writes numbers without words—positional digits or sign-value marks in a fixed scheme. The string 11 is eleven in base ten, three in binary, and two in unary. Roma…
Digit comes from fingers—and base decides how many you needA digit is one symbol in positional notation, like 0–9 in everyday base ten. Latin digiti meant fingers. Binary needs only 0 and 1; hexadecimal borrows letters up to F because six…
The gamma function stretches factorials into complex numbersWritten with a capital Greek gamma, the gamma function is the usual way to extend n! beyond the positive integers. Daniel Bernoulli studied it; defined for all complex numbers exc…
In set theory, a function often is its graphPlot a function of one real variable and you usually get a curve of Cartesian pairs; two variables yield triples that can form a surface. Science and finance use such graphs const…
Hilbert’s twenty-three problems still pace modern mathIn 1900 David Hilbert published twenty-three problems meant to match Henri Poincaré’s stature and set an agenda. A 1902 English Bulletin translation spread the list. Some items no…
Descartes meant “imaginary” as an insult—then Euler wonAn imaginary number is a real multiple of i, where i² = −1; so (5i)² = −25. René Descartes coined the jab in the seventeenth century; Euler and later Gauss made the idea respectab…
The imaginary unit solves x² = −1 when reals cannotDenoted i (or j in electrical engineering), the imaginary unit is a constant satisfying x² = −1. Pairing it with the reals under + and × builds the complex numbers a + bi. Both i …
Inequalities compare sizes without forcing equalityIn mathematics an inequality relates two expressions by a non-equal comparison, usually of size on the number line. Strict forms a < b and a > b forbid equality; they say one quan…
Integers are whole numbers including zero and negativesAn integer is 0, a positive natural number, or the negation of one—so … −2, −1, 0, 1, 2 …. They form a countably infinite set inside the rationals and reals, writable without a fr…
Inverse functions undo a map—only when it is bijectiveThe inverse of f is the function that reverses f’s operation, written f⁻¹ when it exists. Existence demands a bijection: every output hit exactly once. For f(x) = 5x − 7, undo by …
Almost every real number is irrationalIrrationals are reals that are not ratios of integers—π, e, the golden ratio φ, √2 among them. Their decimals never terminate or repeat. Cantor’s contrast of uncountable reals wit…
Napier’s logarithms turned multiplication into additionA logarithm asks what exponent the base needs to make a number—so log₁₀(1000) = 3. John Napier introduced them in 1614 to ease calculation; tables and slide rules followed. Euler …- Euclid's GCD trick still secures internet mathReplace the larger integer by its difference with the smaller—or better, by the remainder after division—and the greatest common divisor stays put. Euclid wrote that down around 3…
The function that equals its own slope is uniqueAmong real functions, only one sends zero to one and matches its derivative at every point: the exponential, written exp(x) or e^x with e about 2.718. It turns addition into multi…- Raising to a power began as repeated multiplicationExponentiation pairs a base b with an exponent n. For positive integers, b^n means multiplying b by itself n times; the idea then stretches to zero, negatives, reals, complex numb…
- Zero factorial equals one by empty-product conventionn! multiplies every positive integer up to n, so 5! is 120. Ancient Jain texts and the Sefer Yetzirah already explored the idea, Plato picked 7!, or 5,040, as the head count for h…
- A formula is a claim; an expression is just a thingScientists pack relations into formulas, mathematical or chemical, so quantities talk to each other in symbols. Expressions name objects; formulas assert equalities or inequalitie…
- Fractions name equal parts—and every rational numberFrom Latin fractus, 'broken', a fraction tells how many equal pieces of a whole you have: one-half, three-quarters. The same a/b notation writes ratios, divisions, and the entire …
Compose functions right to left like nested machinesFunction composition feeds one map's outputs into another's inputs. Write g∘f for 'do f, then g'; associativity lets you drop parentheses, while domains must match, or at least ne…
A function assigns exactly one output to each inputFrom set X to set Y, a function pairs every element of X with precisely one element of Y. Domain and codomain name those sets; f(x) names the value. Graphs of pairs (x,f(x)) make …
Imaginary numbers are as valid as the negatives once wereA complex number has form a + bi, where a and b are real and i satisfies i² = −1. Sixteenth-century mathematicians treated square roots of negatives as symbolic tricks for cubic e…
Checking a trillion cases still does not prove a conjectureA mathematical conjecture is a proposition offered without proof. The Collatz conjecture has been tested past 1.2 trillion without a counterexample, yet one single exception would…- Extra digits after the point signal precision, not vanityDecimal notation uses base ten, now standard worldwide through Hindu–Arabic positional digits. Writing 1.320 milligrams implies tighter measurement bounds than 1.32. Decimals appr…
Division is not commutative, and order changes the answerDivision splits a dividend by a divisor to yield a quotient. Share 20 apples among 4 people and each gets 5; divide 21 among 4 and 1 remains. Unlike multiplication, swapping numbe…
Continuous compounding converges on one mysterious numberThe constant e, roughly 2.71828, underpins natural logarithms and the exponential function that equals its own derivative. Jacob Bernoulli discovered it in 1683 while studying int…- The equals sign sat unused for sixty years after inventionMathematical equality states two quantities share the same value, written A = B. Robert Recorde invented the twin parallel lines in 1557, calling them gemowe lines because nothing…
- Squaring both sides can introduce answers that failAn equation joins two expressions with an equals sign invented by Robert Recorde in 1557. Identities hold for every allowed value; conditional equations hold only for some. Like a…
- Some equations have no algorithm that always worksSolving an equation means finding values that make both sides equal when substituted for unknowns. Solutions may be numbers, functions, or infinite sets. Hilbert's tenth problem w…
Zero arrived in English through Arabic and ItalianZero names an empty quantity and the additive identity: adding it changes nothing, while multiplying by it yields zero. As a digit it holds empty places in decimal writing. Englis…
One is the multiplicative identity, not a primeThe number one is the smallest positive integer and the multiplicative identity: anything times one is itself. By modern convention it counts as neither prime nor composite. Every…
Addition is commutative, associative, and toddler-earlyAddition, marked by plus, is one of arithmetic's four basic operations beside subtraction, multiplication, and division. It combines counts or abstract numbers. Order does not mat…
So-called Arabic digits were shaped in North AfricaThe ten symbols 0–9 that most of the world writes are often called Arabic numerals, also Western digits or Ghubar numerals. Positional decimal ideas are older; these particular sh…- Arithmetic is the shared toolbox under higher mathArithmetic is the branch of mathematics built on adding, subtracting, multiplying and dividing, and more broadly on powers, roots and logarithms. Its name comes from Greek arithmo…
- Leibniz studied binary centuries before silicon needed itBinary notation writes numbers with only two symbols—usually 0 and 1—in base two. Each digit is a bit. Thomas Harriot and later Gottfried Leibniz examined the system in early mode…