Pi represents the ratio of circles circumference to its diameter. It is approximately equal to 3.14. Tau represents the ratio of circles circumference to its radius, which is equal to 2 pi.
Euler's number is another important constant, appearing in many contexts. It is approximately equal to 2.718 and can be defined using this limit or this infinite sum. The imaginary unit, represented as i, has the property that its square is equal to minus 1. In other words, i is equal to the square root of minus 1. The Fibonacci sequence starts with 0 and 1, and each subsequent term is the sum of the two previous terms. If we take the reciprocal of all these terms and sum them, this series converges to a value called the reciprocal Fibonacci constant, which is approximately 3.35. When we look at the ratios of consecutive Fibonacci numbers, this ratio converges to a value called the golden ratio, which is around 1.618.
It is exactly equal to 1 plus the square root of 5 over 2. Imagine a line segment of length A plus B. The ratio of A to B is the same as the ratio of the whole segment to A. Squaring this part gives us the super golden ratio, approximately 1.46.
On the other hand, if the ratio of A to B is equal to the ratio of 2A plus B to A, we get the silver ratio around 2.41, exactly equal to 1 plus the square root of 2. For three numbers connected by this ratio, we have the plastic ratio, approximately 1.32. We also have the golden angle, related to the golden ratio but for angles in a circle. It is approximately 137.5 degrees, or 2.4 radians.
There's also the magic angle, the angle between the diagonal of a cube and any of its three connecting edges. This angle is around 54.7 degrees. Its tangent is exactly equal to the square root of 2. In the Fibonacci sequence we sum the two previous elements.
In the Tribonacci sequence we sum the three previous elements. The ratio between elements in the Fibonacci sequence converges to the golden ratio, but in the Tribonacci sequence it converges to the Tribonacci constant, approximately 1.839. Now, imagine a variation of the Fibonacci sequence where we have a 50% chance of subtracting the previous terms instead of adding them. This creates the random Fibonacci sequence. While this sequence is always different, on average the absolute values of each term grow like powers of 1.13.
This number is known as the Swanlath's constant. If we make the sign of both terms random, the sequence will grow without bound unless we multiply the first term by a factor called beta. If beta is greater than around 0.7, the sequence grows without bound, but if it's smaller, it decays to zero.
This exact value is known as Embry-Trefeathen constant. The diagonal of a square with side length of 1 has a length of the square root of 2, also known as the Pythagoras constant. This number can be written as an infinite continued fraction like this. Now, let's say we take a random number from the range of 0 to 1 and represent it as a continued fraction, like the square root of 2, but with a finite number of coefficients. Locke's constant tells us how much more precision we gain by adding an additional term to the fraction.
It is approximately 0.97, meaning that each additional coefficient in the continued fraction increases the accuracy of the approximation by about 0.97 decimal places. The geometric mean is similar to the arithmetic mean, but instead of adding the numbers and dividing by the count, we multiply the numbers and then take their root. When we represent a random real number as a continued fraction and take the geometric mean of its coefficients, it almost always equals approximately 2.685. This value is known as Kinchin's constant. Numbers for which this is not true include square root of 2, e, and rational numbers.
The harmonic series starts with 1, followed by 1 half, then 1 third, and so on. We can graph the sum of the first x terms of this series. We can also graph the natural logarithm of x.
The distance between the two graphs represents the Euler-Mascaroni constant, which is approximately 0.577. If we sum the reciprocals of only prime numbers, we get a different series. The distance between the graphs of this series and the natural logarithm of the natural logarithm of x is approximately 0.26, known as Meisel-Merton's constant.
Next, let's construct a prime counting function, denoted as pi. This function returns the number of primes up to a given number. For example, pi equals 5, because there are 5 prime numbers up to 12. We can graph this function, and we can approximate it with the function x over ln.
To improve this approximation, we subtract a constant b from the denominator. This constant is called Legendreist constant. In 1808, Adrien Marie Legendre calculated it to be approximately 1.08. However, it is now known that this constant is exactly equal to 1. Euler's number, which we have already seen, is approximately 2.718. The omega constant is a number that satisfies the equation omega times e to the power of omega equals 1. This constant is approximately 0.567.
Gelfand's constant is equal to e raised to the power of pi. which is around 23.14. Interestingly, it is also exactly equal to minus 1 to the power of minus i.
The Goldfond-Schneider constant is equal to 2 raised to the power of the square root of 2. Sylvester's sequence starts with 2. The next term is always the product of all the previous terms plus 1. So the second term is 2 plus 1, the third term is 2 times 3 plus 1, and so on. Now we subtract 1 from all these terms and take the reciprocal. Then we calculate the alternating sum of these terms, adding the first term, subtracting the second, adding the next, and so on. This gives us an infinite sum equal to approximately 0.64, known as Cahen's constant.
Catalan's constant is defined as the alternating sum of the reciprocals of the squares of the odd numbers. Its value is approximately 0.91. A Perry's constant is the sum of the reciprocals of the cubes of all natural numbers, with a value of around 1.2.
The Wallis constant, or Wallis product, is defined as 2 over 1, times 2 over 3, times 4 over 3, times 4 over 5, and so on. The value of this infinite product is exactly equal to pi over 2. The square root of 1 times the square root of 2 times the square root of 3, and so on, has a value of around 1.66, and is called the Sommas Quadratic Recurrence Constant. Let's define a sequence that starts with 1. The next term is the square root of 2. then the cube root of 3, and so on. Now let's calculate the alternating sum of this sequence, starting with the negative first term.
With just one term, the sum is equal to minus 1. With two terms, the sum is approximately 0.41. With three terms, it is minus 1.02. With four terms, it is 0.38.
And so on. This sum oscillates infinitely, but the upper point never goes lower than the MRB constant, which is approximately 0.187. discovered by Marvin Ray Burns.
A pair of twin primes consists of two primes that differ by two, like 3 and 5, or 11 and 13. The sum of the reciprocals of all twin primes is approximately 1.9, known as Brun's constant. A prime quadruplet consists of two pairs of twin primes separated by 4, such as 5 and 7, and 11 and 13. The sum of the reciprocals of all prime quadruplets is approximately 0.87, and is called Brun's constant for prime quadruplets. Every natural number can be written as a product of primes raised to some powers. For example, 504 can be expressed as this product. In this case, the largest exponent is 3. On average, the largest exponent appearing in the prime factorization of any natural number is approximately 1.7 and is known as Niven's constant.
Objects in space usually have orbits in the shape of an ellipse. The Kepler equation helps relate certain properties of these orbits. Imagine object A orbiting around object B.
B in an elliptical orbit. Let's also draw a circle circumscribing this orbit and place a point on the circle. This point travels around the circle in the same time it takes object A to travel around its orbit. This angle is called the mean anomaly, represented by the capital letter M.
Next we draw a vertical line through object A. This angle is called the eccentric anomaly, and is represented by the capital letter E. The small letter E represents the eccentricity of the orbit.
indicating how much the orbit is stretched. A circle which isn't stretched has an eccentricity of zero, while a highly stretched ellipse has an eccentricity closer to one. The solutions to Kepler equation converge only if the eccentricity of an orbit is smaller than approximately 0.66, a value known as the Laplace limit. A parabola can be defined using a point called the focus. We also need a line called the directrix.
The parabola consists of every point that has the same distance to the focus and the directrix. The horizontal line passing through the focus is called the Latus Rectum. The ratio of this distance to this distance is approximately 2.29 and is the same for any parabola. This ratio is known as the universal parabolic constant. Now let's define two focal points at a distance of 2C from each other, and a point P that satisfies the equation where the product of the distance from P to each focus equals C squared lies on the Lemniscate curve.
The Lemniscate constant is the ratio of the perimeter of this curve to its diameter, approximately equal to 2.62, and it remains the same for every Lemniscate. To calculate the arithmetic geometric mean of two numbers, we first find the arithmetic and geometric mean of those numbers and then repeat the process with the new values, continuing until the values converge. The arithmetic geometric mean of 1 and the square root of 2 is approximately 1.19. Its reciprocal, around 0.83, is known as the Gauss's constant.
Interestingly, this constant is exactly equal to the Lemniscate constant divided by pi. The number 89 can be expressed as the sum of two squares. If we have a large number x, the total number of positive integers below x that can be expressed as the sum of two squares is approximately equal to b times x over the square root of ln. Here b is the Landau-Ramanujan constant, which is approximately 0.76. Let's create a sequence called the Look-and-Say sequence.
It starts with 1. To generate the next term, we simply look at the previous term and describe it. We see one digit 1, so the next term is 1 1. In the second term, we see two digits 1, so the next term is 2 1. Next we see one digit 2. and one digit one, so the next term is 1211. This sequence continues indefinitely. On average, after each iteration, the sequence grows by a factor of approximately 1.3.
This growth rate is known as the Conway's constant, and interestingly, it is also the root of this polynomial. The Thumor sequence starts with zero. To generate the next terms, we copy the entire sequence and change all zeros to ones and all ones to zeros. Starting with zero, we copy it, and change it to 1. Next we copy 0, 1 and change it to 1, 0. This process continues indefinitely. If we write the Dewey-Morse sequence as a binary fraction, we obtain the pre-wave Dewey-Morse constant.
In the decimal system, this constant is approximately 0.41. Champernone's constants are a group of constants corresponding to each numerical base. The champernone constant for base 10 is 0.11.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and so on. For binary, it is 0.1, 10, 11, and so on. We can generate a different champer known constant for any base.
The Copeland-Erdos constant is similar to the champer known constant for base 10, but it is constructed using only prime numbers. Now let's create a fraction where if the decimal position is a prime number, the digit is 1, if it's not, the digit is 0. This fraction is written in binary. Its decimal representation is known as the prime constant. Let's define another sequence called the Kolakowski sequence.
In this sequence, we repeatedly alternate between adding 1s and 2s. We start with 1, indicating that we add one digit. Next we add 2, meaning we add two digits.
Continuing, we see the next digit is 2, so we add two digits 1. Then we see a 1, meaning we add one digit 2. This pattern continues indefinitely. If we subtract 1 from each term of the Kolakoski sequence and then represent it as a binary fraction, we obtain the Kolakoski constant. In base 10, this constant is represented like this.
Mill's constant is defined as the smallest positive number that when raised to the power of 3 to the power of n and rounded down results in a prime number for any natural number n. Currently, it is believed that its value is approximately 1.3, although this has not been proven. Using this value for n equal to 1, we get 2, for 2 we get 11, for larger values of n we obtain increasingly large numbers.
With larger and larger n, it becomes extremely difficult to verify if the resulting number is prime. The factorial of a number is defined as the product of all natural numbers up to that number. This means we can only calculate factorials of natural numbers. To graph the factorial function, we can plot the integer points.
We can extend the factorial function to any number using the gamma function. We can also graph the reciprocal of the gamma function. The area under the graph of this function to the right of the y-axis is approximately 2.8, and is known as the Franson-Robinson constant. Let's draw a circle with a radius of 1. Next circumscribe an equilateral triangle. Then again circumscribe a circle.
Then a square. And another circle. Then a regular pentagon. And so on.
The radius of the circle will never exceed approximately 8.7, a value known as the polygon circumscribing constant. Now, let's draw another circle with a radius of 1, but this time inscribe a triangle inside it. We continue the pattern as previously.
The radius of the inscribed circle will never be smaller than approximately 0.11, which is known as the Kepler-Buchamp constant. Interestingly, this constant is exactly equal to 1 divided by the polygon circumscribing constant. The dotty number is the solution to this equation.
Its value is approximately 0.73. Let's define a sequence where each element is equal to 1 plus 1 over the previous element raised to the power of its position in the sequence. For example, if we start with 1, the next element is 2 to the power of 1. The next element is 1.5 to the power of 2, and so on. This sequence quickly begins to grow and oscillate between large numbers and 1. If we start with 2, the sequence still jumps up and down.
Starting with 1.5 produces a similar pattern. This behavior occurs for any starting number, except when starting with the Fourier's constant, which is around 1.18. If we start with this number, the sequence grows like this.
A Mersenne number is a number of the form 2 to the power of n minus 1. The sum of the reciprocals of all Mersenne numbers is approximately 1.6, and is known as the Erdős-Borvene constant. Mersenne numbers are also used to find very large prime numbers, some with millions of digits. To learn how these giant prime numbers are discovered, watch this video next.