On our discussions page, folks have been wondering how many others are using Brilliant? Currently, there are 10,000 monthly active users, and 40,000 registered users on Brilliant.org. We are steadily growing, and will keep you updated in the coming weeks and months as the community evolves. From the graph above, you will all notice an impressive international distribution of users. In the coming weeks, we will be exploring ways to permit all of you to better meet and benefit from the diverse collection of avid thinkers on our site. If you have friends, family, teachers, or meet a stranger who you think would enjoy Brilliant, spread the word. The more the merrier!

Graph of the inequality in worked example number 1

[This is targeted at a Level 2 student. Unless otherwise mentioned, all quantities are real numbers.]

In an algebra course, students are introduced to the concept of order on the real numbers. This is a relation between the real numbers, denoted by the symbol $>$, which is often read as “greater than”. We have the following 4 order axioms for real numbers:

1. [Trichotomy] If $a, b$ are real numbers, then one and only one of the following is true: $a >b$, $a = b$, $b > a$.
2. [Transitivity] If $a, b, c$ are real numbers and $a > b , b > c$, then $a > c$.
3. If $a, b, c$ are real numbers and $a>b$, then $a + c > b + c$.
4. If $a, b, c$ are real numbers such that $a>b, c > 0$, then $ac > bc$.

Though named after Blaise Pascal(1623-1662), the triangular array of binomial coefficients had been studied for centuries before him by Indian, Arabic, Chinese, German, and Italian mathematicians. The above drawing is a 14th century Chinese depiction of Pascal’s Triangle attributed to the mathematician Yang Hui in 1303.

[This is targeted at a Level 2 student.]

In the blog post about Combinations, we introduced the concept of binomial coefficients. We showed that the binomial coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ objects from a set of size $n$. Binomial coefficients also arise naturally when raising binomial expressions to powers, as seen in the Binomial Theorem:

As you progress further into college math and physics, no matter where you turn, you will repeatedly run into the name Gauss. Johann Carl Friedrich Gauss is one of the most influential mathematicians in history. Gauss was born on April 30th 1777  in a small German city north of the Harz mountains named Braunschweig. The son of peasant parents(both were illiterate), he developed a staggering number of important ideas and had many more named after him. Many have referred to him as the princeps mathematicorum, or  the “Prince of Mathematics.”

Patterns are visually ubiquitous, both in the forms of nature and as motifs in our art and architecture. Much of scientific and mathematical thought is spent discovering what principles cause patterns and generate their predictability. In math, patterns are often much harder to spot than they are in mosaic.

[This post is targeted at a level 1 student.]

When solving math problems, it is often good if we are able to recognize a certain pattern in the numbers that we are seeing. This will allow us to hypothesize what the general term of the sequence will look like, which could help to guide the approach for the problem.

You should be aware of sequences like the integers, odd numbers, perfect squares, primes, factorials, exponents, etc, and these should be easily identified.

The distribution of Gaussian Primes with norms up to 500.

[This post is targeted at a Level 4 student. You should have read Gaussian Integers. This post is devoted to the classification of Gaussian primes. We will prove the unique factorization theorem for the Gaussian integers in the next post.]

Perhaps the most useful concept in Gaussian integers is the norm: $N(a+bi)=a^2+b^2$. As discussed in the first post, it is always a non-negative integer, and it is multiplicative: the norm of the product is the product of the norms. One application of this is the classification of Gaussian units.

There is no common fraction that will divide evenly into 1 and the square root of two. This was initially distressing to Ancient Greek mathematicians.

The first man to recognize the existence of irrational numbers might have died for his discovery. Hippassus of Metapontum was an Ancient Greek philosopher of the Pythagorean school of thought. Supposedly, he tried to use his teacher’s most famous theorem $a^{2}+b^{2}= c^{2}$ to find the length of the diagonal of a unit square. This revealed that a square’s sides are incommensurable with its diagonal, and that this length cannot be expressed as the ratio of two integers. The other Pythagoreans believed dogmatically, that only positive rational numbers could exist. They were so horrified by the idea of incommensurability, that they threw Hippassus overboard on a sea voyage, and vowed to keep the existence of irrational numbers an official secret of their sect.