50 Fun Facts About Algebra

The word algebra comes from the Arabic al-jabr, a term for restoring that was also used for which medical treatment?

Bonesetting was called algebra in English in the 15th and 16th centuries, a medical usage that probably came from Arab doctors in Spain.

Welsh mathematician Robert Recorde invented which symbol in 1557 because he was tired of writing the same words?

The equals sign made its debut in The Whetstone of Witte, where Recorde chose two parallel lines of one length because no two things can be more equal.

Which 9th-century Persian scholar wrote the book that named algebra and later gave his own name to the word algorithm?

Al-Khwarizmi worked at the House of Wisdom in Baghdad around 820, and Latin translators later rendered his name as Algoritmi.

Which English shoemaker's son gives his name to the algebra of true and false values behind computer logic?

George Boole set out the system in The Laws of Thought in 1854, and Claude Shannon later applied it to relay circuits, the forerunners of digital computers.

The graph of a quadratic function such as y = x² is which U-shaped curve, also traced by a thrown ball?

A parabola also gives satellite dishes and car headlight reflectors their shape, because the curve gathers parallel rays at a single focal point.

In the straight-line equation y = mx + b, the letter m stands for which measure, often described as rise over run?

Slope has been written as m since at least 1844, when the letter first appears in an English textbook, yet nobody is sure why it was chosen.

Which Italian, famous for a sequence about breeding rabbits, spread Arabic algebra in Europe with his book Liber Abaci?

Fibonacci was educated in what is now Algeria, where his father directed a trading post, and it was there that he learned the Hindu-Arabic numerals.

Following the order of operations taught as PEMDAS, what is the value of the expression 1 + 2 × 3?

The value is 7, because multiplication is done before addition. Working strictly left to right gives 9, the slip the PEMDAS mnemonic is meant to prevent.

The classroom mnemonic FOIL, short for first, outer, inner, last, is a method for multiplying two of what?

Binomials are expressions with two terms, such as x + 3, so the method gives four products. FOIL is so familiar in American schools that it is used as a verb.

Solving x² = −1 needs which kind of number, made of a real part and a part once mocked as imaginary?

Complex numbers such as 3 + 2i were long written off as fictitious and useless, yet they are now everyday tools in electromagnetism and quantum mechanics.

Which British mathematician proved Fermat's Last Theorem, on the equation aⁿ + bⁿ = cⁿ, after years of secret work?

Andrew Wiles unveiled his proof in 1993, some 350 years after Fermat claimed one too long for a book's margin, then had to repair an error before it was accepted.

Which ancient people left clay tablets, nearly 4,000 years old, that solve quadratic problems in cuneiform script?

The Babylonians counted in base 60, and one of their tablets gives the square root of 2 correct to about six decimal places.

The numbers produced by multiplying out (a + b)ⁿ fill the rows of which pattern, studied in China by Yang Hui?

Pascal's triangle builds each entry by adding the two numbers above it, and it was studied in India, Persia and China centuries before Blaise Pascal wrote about it.

What name is given to a rectangular array of numbers in rows and columns, from a Latin word for womb?

Matrix was coined for the purpose by James Joseph Sylvester in 1850, who pictured the array as a parent giving birth to the values called determinants.

Which French philosopher's book La Géométrie began the habit of using x, y and z for unknown quantities?

René Descartes also pioneered the raised numbers used for powers, such as the small 2 in x², while reserving a, b and c for known quantities.

Which property of addition, whose name comes from a word for exchanging, guarantees that a + b always equals b + a?

Commutative operations ignore order, which is why 3 + 5 and 5 + 3 match. Subtraction and division fail the test, since 3 − 5 is not 5 − 3.

In the algebraic term 5x, what name is given to the 5, the number written in front that multiplies the letter?

Coefficient is the name for that multiplier, and when none is written, as in x², it is understood to be 1.

Using the distributive property to remove the parentheses, 3(x + 5) expands to which expression?

The result is 3x + 15, because the 3 outside multiplies every term inside the parentheses. Stopping at 3x + 5 is a common slip.

What value of x solves the two-step equation 2x + 5 = 17, where the unknown is doubled and then increased?

The answer is 6: taking 5 from both sides leaves 2x = 12, and halving gives x. Treating both sides alike at every step keeps the equation balanced.

The two vertical bars in |−8| = 8 denote which quantity, a number's distance from zero on the number line?

Absolute value strips away the sign, so −8 and 8 both give 8. The bar notation was introduced by Karl Weierstrass in 1841.

Under the associative property, (2 + 3) + 4 equals 2 + (3 + 4); which feature of the sum has changed?

The grouping is all that moves: the parentheses shift while the numbers stay in line. Subtraction lacks the property, since (10 − 5) − 3 is 2 but 10 − (5 − 3) is 8.

Zero is called the additive what, because adding it to any number leaves that number unchanged?

Identity is the title, and multiplication has one too: the number 1, since multiplying by 1 changes nothing.

Which name is given to a number's multiplicative inverse, such as 1/5 for 5, since the pair multiply to 1?

Reciprocal is the everyday name, and it explains the rule for dividing by a fraction: flip the fraction and multiply instead.

If a = b and b = c, then a = c: which property of equality appears in Euclid's Elements?

Transitive reasoning links a chain of equal quantities end to end. Euclid put it as things equal to the same thing being equal to one another.

Combining like terms, which share the same letter and power, what does 3x + 5x + 2 simplify to?

It becomes 8x + 2, because 3x and 5x count the same thing and can be added, while the lone 2 has no partner and stays as it is.

Which algebraic expression matches the phrase five less than twice a number n, where the subtraction comes last?

It is 2n − 5: twice the number comes first, then 5 is taken away. Writing 5 − 2n follows the word order but subtracts the wrong way round.

What is the value of 2x² when x = 3, given that the power is worked out before the multiplication?

The value is 18, because the 3 is squared to give 9 before being doubled. Doubling first and then squaring gives 36, which is the value of (2x)² instead.

By the product rule for exponents, x² × x³ simplifies to which single power of x?

The answer is x⁵, since x² is two copies of x and x³ is three more, five in all. Exponents multiply only when a power is itself raised to a power.

Powers of 2 halve as the exponent falls, from 8 to 4 to 2, so 2 to the power 0 equals what?

It equals 1, and the same holds for every non-zero base. The convention keeps the laws of exponents consistent, since dividing any power by itself must give 1.

The pattern a² − b² = (a − b)(a + b), a shortcut for mental math, is known by which name?

The difference of squares turns 21 × 19 into (20 + 1)(20 − 1), which is 400 − 1 = 399, with no long multiplication needed.

What is the degree of the polynomial 5x⁴ + 2x − 1, which makes it a quartic rather than a cubic?

The degree is 4, the highest power of x that appears; the 5 in front is only a multiplier. Degree 2 is called quadratic and degree 3 cubic.

Which factored form multiplies out to x² + 5x + 6, built from two numbers that add to 5?

The answer is (x + 2)(x + 3), because 2 and 3 add to 5 and multiply to 6. Hunting for such a pair is the standard first step in factoring a quadratic.

Which two values of x solve the factored equation (x − 2)(x + 5) = 0, each making one factor vanish?

The solutions are 2 and −5, since a product can only be zero when one of its factors is zero. The signs flip because x − 2 vanishes at 2, not at −2.

Dividing both sides of −2x > 6 by −2 gives which solution, under the rule for negative divisors?

The solution is x < −3, because dividing or multiplying an inequality by a negative number reverses its direction. Try x = −4: −2 × −4 is 8, which beats 6.

Adding the equations x + y = 20 and x − y = 4 cancels y and reveals which value of x?

The value is 12: adding the two equations gives 2x = 24, and y must then be 8. Cancelling one unknown this way is called elimination.

In the quadratic formula, what is the expression b² − 4ac called, whose sign reveals how many real solutions exist?

The discriminant signals two real solutions when positive, exactly one when it equals zero, and none on the real number line when negative.

The set of every input a function will accept is called its what, the counterpart of the range of outputs?

Domain limits matter in practice: 1/x must leave out 0, and the real square root must leave out every negative input.

A sequence such as 5, 8, 11, 14, which grows by the same amount at every step, is called what?

Arithmetic sequences plot as dots along a straight line, so they are also called linear sequences. A geometric one multiplies by a fixed number at each step instead.

Three consecutive whole numbers add up to 33; calling the smallest one n, which number is n?

It is 10, since n + (n + 1) + (n + 2) = 33 simplifies to 3n + 3 = 33. The middle number, 11, is simply 33 divided by 3.

If a function is defined by f(x) = x² − 1, what is f(5), read aloud as f of five?

The value is 24, found by swapping every x for 5 to get 25 − 1. The parentheses do not mean multiplication here: f(5) is the output when the input is 5.

Which mathematician of ancient Alexandria wrote the Arithmetica and is sometimes called the father of algebra?

Diophantus filled the Arithmetica with number puzzles solved by equations, and it was in the margin of a copy that Fermat later scribbled his famous Last Theorem.

Which German 'Prince of Mathematicians' devoted his doctoral thesis to the fundamental theorem of algebra?

Carl Friedrich Gauss earned his doctorate with it in 1799 and returned to the theorem with three more proofs, the last of them fifty years later.

Which mathematician, mourned by Einstein in a New York Times letter, reshaped abstract algebra with her theory of rings?

Emmy Noether spent four years lecturing at Göttingen under David Hilbert's name, because the faculty objected to a woman holding the post.

Which French founder of group theory in algebra died at twenty from wounds received in a duel?

Évariste Galois left his ideas in a letter to his friend Auguste Chevalier, and their value was recognised only a decade later, when Joseph Liouville studied his papers.

Which Swiss mathematician was the first to write f(x) for a function and also popularized π as a symbol?

Leonhard Euler also introduced e for the base of natural logarithms, i for the square root of −1 and the Greek letter Σ for sums.

Which Persian poet and astronomer wrote an algebra treatise solving cubic equations with intersecting conic sections?

Omar Khayyam is best known in the West for the Rubaiyat, yet he also designed a solar calendar of remarkable accuracy.

Abel and Ruffini showed that a general formula in radicals first becomes impossible for which type of equation?

Quintic equations, those of degree five, broke a long run of success: formulas for degrees two, three and four had all been found by the mid-1500s.

Which Italian physician and gambler published the solution of the cubic equation in his book Ars Magna?

Gerolamo Cardano had learned the method from Tartaglia, who claimed Cardano had sworn to keep it secret, and the two feuded over it for a decade.

The classic error nicknamed the freshman's dream wrongly claims that (a + b)² equals which expression?

It claims a² + b², dropping the middle term 2ab that the true expansion contains. Squaring 3 + 4 shows the gap: 49 on one side against 25 on the other.

Pulling out the perfect square 25 simplifies the square root of 50 to which exact form?

It becomes 5√2, because 50 is 25 × 2 and the square root of 25 is exactly 5. The same trick turns the square root of 12 into 2√3.