50 Fun Facts About Precalculus

Young Carl Friedrich Gauss is said to have quickly summed the whole numbers 1 to 100; what total did he reach?

5,050 is the total: pairing 1 with 100, 2 with 99 and so on gives 50 pairs that each add up to 101. The same trick sums any arithmetic series.

The mnemonic All Students Take Calculus helps trigonometry students recall which fact about the six trig functions?

Their signs in each quadrant are what the initials track: all are positive in the first quadrant, only sine in the second, tangent in the third and cosine in the fourth.

Johannes Kepler's first law says each planet travels around the Sun along which curve rather than a perfect circle?

An ellipse, a stretched circle, is the path; Kepler published the law in 1609 after fitting Tycho Brahe's careful observations of Mars.

Which French philosopher, living in the Netherlands at the time, gives the Cartesian coordinate plane its name?

René Descartes set out the idea in 1637, letting algebra describe geometry; Pierre de Fermat found it independently but never published.

One radian is the angle at a circle's center whose arc is exactly as long as which part of the circle?

The radius sets the size of the unit, so one radian is the same angle on a coin as on a planet's orbit: about 57.3 degrees.

Which famous list, starting 0, 1, 1, 2, 3, 5, 8, makes each new term the sum of the previous two?

The Fibonacci sequence reached Europe in a 1202 book by Leonardo of Pisa, who used it to model a growing population of rabbits.

The identity sin²θ + cos²θ = 1 is named after which ancient Greek mathematician?

Pythagoras gets the credit because the identity is his theorem in disguise: sin θ and cos θ are the two legs of a right triangle whose longest side measures 1.

Which capital Greek letter signals adding up the terms of a series in mathematical notation?

Sigma stands for sum; capital pi does the same job for multiplication, signaling the product of a run of terms.

Jacob Bernoulli met the constant e while studying compound interest; e is approximately equal to which value?

2.718 is e to three decimal places: it is what $1 grows to in a year at 100% interest when the interest is compounded continuously.

A full 360° turn around a circle equals which angle in radians, roughly 6.28?

2π radians make a full turn, because a circle's circumference is 2π times as long as the line from its center to its edge.

Which angle is coterminal with 45°, ending on the same terminal side after one extra full turn?

405° is 45° plus one 360° rotation, so both angles point the same way; subtracting a turn instead gives −315°, which is coterminal too.

Which positive angle is coterminal with −60°, found by adding one full 360° turn to the negative angle?

300° lands on the same terminal side as −60° because the two differ by exactly one full turn: 60° clockwise and 300° counterclockwise finish in the same place.

Which trigonometric function is the reciprocal of sine, so it is undefined wherever sin θ equals zero?

Cosecant is 1 over sine, a pairing that trips students up because the co-function goes with sine while secant goes with cosine.

What is the exact value of sin 30°, the shortest side over the longest in a 30-60-90 triangle?

1/2 is the value because a 30-60-90 triangle is half of an equilateral triangle, so its shortest side is half its longest.

The graph of y = tan x repeats its pattern after which interval, half as long as the sine wave's cycle?

180° is the period of tangent, because opposite points on a circle give the same slope; sine and cosine need a full 360° before they repeat.

In y = 3 sin x, the 3 sets the wave's peak height above the x-axis; what is that height called?

Amplitude is the peak height of a wave; in a sound wave it relates to the volume, while frequency sets the pitch.

Which name is given to the curve x² + y² = 1, whose points are (cos θ, sin θ)?

The unit circle lets sine and cosine take values for any angle, not just the acute angles that fit inside a right triangle.

Which law extends the right-triangle rule a² + b² = c² to any triangle, using two sides and the included angle?

The law of cosines adds a correction term, −2ab cos C, to the familiar formula; when C is 90° that term vanishes and the right-triangle rule returns.

Which kind of angle, the acute angle between a terminal side and the x-axis, helps find trig values in any quadrant?

Reference angles let one small table do all the work: 150°, 210° and 330° all share the reference angle 30°, so their trig values match up to sign.

In the arithmetic sequence 5, 7, 9, 11, 13, the constant 2 added at every step is called what?

The common difference can be negative too: 10, 7, 4, 1 is still an arithmetic sequence, with a common difference of −3.

Each term of the geometric sequence 2, 6, 18, 54 is three times the one before; which term comes next?

162 follows, three times 54. Growth by a fixed multiplier soon outruns growth by a fixed step, which is why this pattern describes compound interest.

The factorial 5!, the product of every whole number from 5 down to 1, equals which number?

120 is 5 × 4 × 3 × 2 × 1, and it is also the number of different orders in which five books can be lined up on a shelf.

Zeno's walker covers half the distance, then a quarter, then an eighth, forever; those fractions add up to which number?

1 is the total, which answers Zeno: the walker's endless list of ever-shorter steps still covers a finite distance.

Which number array, named in the West after a French mathematician, lists binomial coefficients row by row?

Pascal's triangle was studied centuries before him in India, Persia and China; in Iran it is known as Khayyam's triangle, after Omar Khayyam.

Expanding (x + y)² gives x² plus y² plus which middle term, the one that beginners often leave out?

2xy is the term; writing (x + y)² = x² + y² without it is such a common slip that it is nicknamed the freshman's dream.

Which series adds the reciprocals of the counting numbers and grows without bound though its terms shrink toward zero?

The harmonic series eventually passes every number, a fact Nicole Oresme proved in the 14th century, long before calculus existed.

Adding the first few odd numbers, as in 1 + 3 + 5 + 7, always produces which kind of total?

A perfect square appears every time: 1 + 3 + 5 + 7 = 16, because each new odd number wraps an L-shaped border around the square before it.

A vector runs 3 units right and 4 units up, written 3i + 4j; which magnitude, or arrow length, does that give?

5 is the length, since 3² + 4² = 25 and the square root of 25 is 5; the 3-4-5 right triangle is the best-known one with whole-number sides.

Speed and temperature have a size but no direction; what is such a quantity called, in contrast to a vector?

A scalar is fully described by one number, which is why speed is a scalar while velocity, which also needs a direction, is a vector.

Two nonzero vectors that meet at a right angle, such as i and j, always have which dot product?

0 is the dot product of any perpendicular pair, which gives a quick test for right angles: multiply matching components, add, and see if the total vanishes.

Adding vectors component by component, what is the sum of i + 2j and 3i + 4j?

4i + 6j is the sum: the i parts add to 4 and the j parts to 6, the same as walking along one arrow and then along the other.

The word vector comes from a Latin word with which meaning, because a vector takes one point to another?

Carrier is the meaning: a vector carries a point from A to B, and the word shares its Latin root, vehere, with vehicle.

Which Irish mathematician, the inventor of quaternions, introduced the term vector for their directed part?

William Rowan Hamilton had the idea on a walk in Dublin in 1843 and carved the key equation into the stone of Broom Bridge with his penknife.

For f(x) = 1/x, the set of allowed inputs, every real number except 0, is called the function's what?

The domain is where a function is allowed to operate; two classic restrictions are no dividing by zero and no square roots of negative numbers.

Which line test, named for the direction of its lines, tells whether a graph such as a circle is a function?

The vertical line test works because a function gives one output per input; a circle fails it, since most upright lines cut it at two heights.

The graphs of y = eˣ and its inverse y = ln x are mirror images of each other across which line?

y = x is the mirror, because an inverse function swaps every input with its output, turning each point (a, b) into (b, a).

The curve y = 1/x gets ever closer to the x-axis without reaching it; what is such a line called?

Asymptote comes from Greek for not falling together, a fair description of a curve and a line that keep closing in on each other.

By the remainder theorem, dividing a polynomial f(x) by x − 2 leaves a remainder equal to which value?

f(2) is the remainder, so no long division is needed; and when f(2) equals zero, x − 2 divides the polynomial exactly, which is the factor theorem.

Every point of a parabola lies equally far from its focus and from which fixed line outside the curve?

The directrix never touches the parabola; letting the two distances keep a fixed ratio other than 1 produces the other conic sections instead.

The base-10 logarithm of 1,000 is which number, because 10 raised to that power gives 1,000?

3 is the power needed, since 10³ = 1,000; for 10, 100, 1,000 and so on, the base-10 logarithm simply counts the zeros.

Which Scottish mathematician introduced logarithms, a shortcut that turns long multiplications into simple additions?

John Napier published them in 1614 in a book whose title translates as Description of the Wonderful Canon of Logarithms.

Which Swiss mathematician wrote Introductio in analysin infinitorum, a 1748 text seen as the first precalculus book?

Leonhard Euler opened the book with variables and functions, and presented the logarithm as the inverse of an exponential function, the way it is still taught.

The imaginary unit i, which engineers write as j, is defined by having which number as its square?

−1 is the square of i; engineers switch to j because the letter i already stands for electric current.

In the quadratic formula, the expression b² − 4ac under the square root sign goes by which name?

The discriminant sorts quadratics at a glance: positive means two real solutions, zero means one repeated solution and negative means none on the number line.

Which number measures how far a conic section departs from circular, being 0 for a circle and 1 for a parabola?

Eccentricity above 1 gives a hyperbola, and the scale has no upper end: the larger the number, the more the curve opens out toward a pair of straight lines.

Which ancient Greek from Perga wrote the eight-book Conics around 200 BC and gave the parabola its name?

Apollonius also named the hyperbola, and every curve in his Conics comes from slicing a cone with a flat plane at a different angle.

Which idea, the value a function approaches as its input nears a point, forms the bridge from precalculus to calculus?

The limit is what turns an average rate over a shrinking interval into an exact one; derivatives and integrals are both defined with it.

Which polar curve is traced by a point on a circle rolling around an equal circle?

Cardioid is the name, though the curve looks more like the cross-section of an apple; a common microphone pickup pattern has the same shape.

Adding two vectors tip to tail gives one arrow from the first tail to the last head; what is it called?

The resultant works for forces too: several pushes and pulls on one object have the same effect as their single resultant force.

Which hand-held calculating tool, built on logarithmic scales, served engineers until pocket calculators arrived?

The slide rule was developed in the 17th century by William Oughtred and others; it faded quickly once the HP-35 scientific calculator arrived in 1972.