Numbers and Math

Zero is the only number that cannot be used as a divisor in ordinary arithmetic without breaking the rules.

Prime numbers have exactly two factors, which is why one is not considered prime by mathematicians.

Two is the only even prime number, since every other even number can be divided by two.

Euclid proved more than two thousand years ago that the list of prime numbers never comes to an end.

Twin primes are pairs of primes that differ by two, such as eleven and thirteen or seventeen and nineteen.

Negative numbers were once rejected by mathematicians before becoming standard tools for debt, temperature, and direction.

The number pi is irrational, meaning its decimal expansion never ends and never settles into a repeating pattern.

Pi has been calculated to trillions of digits, though only a few dozen are needed for practical engineering.

March fourteenth is celebrated as Pi Day because the written date looks like the first digits of pi.

Euler's number e appears naturally in compound interest, population growth, and many problems involving continuous change.

The golden ratio is roughly one point six one eight and shows up in spirals, art, and plant growth.

Fibonacci numbers run one, one, two, three, five, and onward, with each term the sum of the two before it.

Dividing consecutive Fibonacci numbers gives values that approach the golden ratio more and more closely.

The square root of two was one of the first numbers proven irrational, which unsettled ancient Greek mathematicians.

Transcendental numbers like pi and e are not roots of any polynomial with whole number coefficients, and almost all real numbers are transcendental.

Imaginary numbers were introduced to give meaning to the square root of negative one, written as i.

Complex numbers combine a real part and an imaginary part, and they are essential in electrical engineering and physics.

A googol is the number one followed by one hundred zeros, far larger than anything in daily experience.

A googolplex is a one followed by a googol zeros, a number too large to write out in full.

Graham's number is so vast that the observable universe lacks the space needed to write all its digits.

Infinity is not a number in the usual sense, and different infinities can actually be different sizes.

A hotel with infinitely many rooms can still take new guests by moving each occupant to the next room.

Zero was used as a placeholder in ancient counting systems long before it was treated as a number itself.

Roman numerals have no zero and no place value, which makes long multiplication awkward compared with modern notation.

Our familiar digits are called Arabic numerals, though they were developed in India before spreading westward.

Place value means a digit position changes its worth, which is why three means different things in thirteen and thirty.

Binary uses only zero and one, and it is the counting system at the heart of modern computers.

A single byte holds eight bits, which allows two hundred fifty-six different values to be represented.

Hexadecimal uses sixteen symbols per place, mixing the usual digits with letters from a through f.

The Babylonians counted in base sixty, which is why hours still have sixty minutes and circles have degrees.

A dozen is twelve, a gross is one hundred forty-four, and both remain common in wholesale counting.

Most people use base ten for counting, likely because humans have ten fingers on their two hands.

Triangles always have interior angles that add to one hundred eighty degrees in flat Euclidean geometry.

The angles of a triangle drawn on a sphere add to more than one hundred eighty degrees, unlike flat triangles.

Parallel lines never meet in flat geometry, but on a curved surface they can eventually converge.

The Pythagorean theorem says the square of the hypotenuse equals the sum of the squares of the other sides.

A three, four, five triangle is the simplest example of a right triangle with whole number side lengths.

The number six is perfect because its proper divisors of one, two, and three add up to six.

Amicable numbers come in pairs where the proper divisors of each sum to the other, as with two hundred twenty and two hundred eighty-four.

A magic square arranges numbers so that every row, column, and diagonal adds to the same total.

The birthday problem shows that a room of roughly twenty-three people has better than even odds of a shared birthday.

In the classic three-door puzzle, switching your choice after one door is opened wins two times out of three.

A coin has no memory, so a long run of heads does not make tails more likely on the next flip.

Expected value multiplies each outcome by its probability, which is how casinos and insurers set long-term prices.

The law of large numbers says averages settle down as more trials are added to an experiment.

Averages can mislead, since the mean, median, and mode answer three slightly different questions about the same data.

Percentages can be tricky, because a fifty percent rise followed by a fifty percent fall leaves you below the start.

The four color theorem states that any flat map needs no more than four colors so neighboring regions always differ.

Euler showed that a network can be traced without lifting a pen when every junction has an even number of connections.

A Mobius strip has only one side and one edge, which surprises anyone who first traces a line along it.

Knot theory studies closed loops that cannot be untangled without cutting, and it has surprising uses in chemistry.

Fractals repeat the same pattern at every scale, which is why coastlines look similarly jagged from far away.