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What Does the Backwards 3 Symbol Mean?

With so many signs and symbols in mathematics, it is sometimes hard to decipher or even understand what each one stands for. For instance, the backward 3 symbol (ε) — what does it mean, and how do mathematicians use it in equations? 

The ε symbol, also known as epsilon, represents the closest number to zero, yet it is not zero. It is not a constant number, and it is variable depending on the equation. You will find it in many fields of mathematics but most commonly in calculus and algebra. While the ε is a variable, due to its changing nature, it is challenging for anyone to define.

Welcome to the world of mathematics, where signs and symbols dominate a student’s examination paper and take over a scientist or mathematician’s work journal. For those of you who thrive in the world of numbers, let’s embark on this journey of understanding the epsilon together.

History of Epsilon

It would come as no surprise that the epsilon symbol originated from Greece. Greece has a rich history with numerous contributions to modern society, and the epsilon symbol is one of them (source).

Epsilon is part of the Phoenician alphabet that the Greeks co-opted from the original symbol when they created their own alphabet. It still resembles the original Phoenician letter today when written in the lower-case form.

In its upper-case form, it looks like a large E. Like the English alphabet, it is the fifth letter of the Greek alphabet and it creates a similar phonetic sound. As it is the fifth letter, many associate it with the number 5.

Initially, they called it “ei,” naming it after the sound that it made. In the Middle Ages, European scholars named it “epsilon” to distinguish it from similar-sounding letters. 

With such a long history, it is understandable that epsilon plays a major role in mathematics, astrology, and science, among other fields. 

How to Use Epsilon

Paul Erdös, a mathematician, may have used the best analogy to describe epsilon’s function — he referred to epsilon as his “children,” and the metaphor is helpful to those of us who are not gifted in math.

When you think of children, you may be thinking of delightful little bundles of energy running around a garden or, perhaps, some demon spawn that throws things at your dog from over the fence. 

Just like children, the epsilon never remains the same for long — and you will likely get a headache trying to understand both children and the epsilon — the larger they grow, the further away from you they get.

As mentioned previously, epsilon is a small number, almost as small as zero, but never achieving pure nothingness like zero.

For those long-suffering algebra students, epsilon is a variable similar to X or Y that they’ve likely used in algebraic equations.

Epsilon is both a real number and a positive number, though it can be ridiculously small or large. Epsilon is a jack of all trades and can act as a stand-in for a wide range of numbers.

Real Numbers

“Real numbers” is a fancy way of saying all numbers. In English, we talk about the lexis (or words) of a language. In mathematics, we talk about the real numbers (again, just plain numbers) we use.

The idea of real numbers is an umbrella term and can cover a variety of number types. By definition, real numbers are any numbers that you can plot on a number line (source).

This includes rational and irrational numbers, fractions, positive and negative numbers, natural numbers, and whole numbers.

Hence, it makes perfect sense that epsilon is an example of a real number. Since epsilon does not have a set value but could be a range of numbers, you can classify it as a real number.

Image by Jeshoots via Unsplash

Positive Numbers

Positive numbers refer to any number greater than zero (source). On a number line, positive numbers appear to the right of zero, while negative numbers appear to the left.

You can quantify everything countable with positive numbers. You also use positive numbers to measure space, volume, distance, and even force and energy. Essentially, anything we can quantify as a number above zero is a positive number.

Since epsilon refers to a number greater than zero but which is not zero, it is a perfect example of a multitude of positive numbers.

If you find yourself confused with how to reference the number zero, check out “Zeroes or Zeros: Understanding the Noun, Verb, and Adjective” to know which one to use when you cannot use the number itself.

What Is the Value of Epsilon?

As mentioned previously, epsilon can refer to any range of numbers that are not zero, but its most general usage is as a stand-in for a number between 0 and 1.

Instead of using the specific fraction or decimal to express an exact number, you can use epsilon.

The difficulty in defining epsilon comes from the fluctuating quality of the “number.” Since epsilon is supposed to refer to the smallest number which is closest to zero, this can be an infinite amount of numbers.

Epsilon can be 0.1 and 0.01 and 0.001, and so on. There will always be a number smaller than the fraction that you can write, and that is why epsilon is such a handy symbol.

It is a representation of a number that is too difficult or cumbersome to write.

Why Do We Need Epsilon?

Epsilon is the Plato, Aristotle, and Socrates of mathematics. You will not often find a use for it in day-to-day mathematical functions, but it does provide a solid foundation for understanding complicated and abstract concepts.

Epsilon-Delta Proof

The most common usage of epsilon is the more easily understood stand-in for a small number, but that is not its only purpose.

Within mathematics, there is the concept of limits. A limit is also known as epsilon-delta due to the usage of the ϵ and δ symbols (source). 

Epsilon-delta is used to evaluate the limits of a function. It requires some breaking down to understand, so let’s start by taking a look at the graph below.

Image by Tahira Hanif

Functions

Within mathematics, a function refers to a process in which there is an input, a relationship that takes place, and an output.

Functions require x and y as part of the equation, and they are usually expressed on a Cartesian plane (drawn in black above).

In an equation, you will generally present functions as f. While x usually stands alone in an equation, you can express y as f(x), which is drawn in red above.

For example, x multiplied by two is an example of a function. X is the input, x 2 is the relationship, and then the output changes depending on the input. If x is equal to 20, then the output is 40. Similarly, if the input is 269, then the output is 538.

Functions are not only restricted to multiplication but rather all forms of mathematical operations such as addition, subtraction, division, and so on. 

Limits

Limits in mathematics means that you are approaching a specific x coordinate on a function line. This means that something is approaching along the f(x) line but never hitting the x coordinate, shown in the limit.

Ordinarily, when evaluating the coordinates of a specific point on f(x), you would substitute the x in the function with a value and determine what the f(x) output would be, but you cannot do this when there is a discontinuity or break in the graph. 

In that case, you would need to reference limits to determine where the discontinuity is.

Epsilon is an example of a limit because it approaches zero but never becomes small enough to be zero. When you present epsilon-delta as an equation, you would write this as:

lim f(x) = L
x→a

F(x) is a function of x, a is the x-value of the point being approached, and L is the y-value being approached.

The graph above is discontinuous at the coordinate where a and L would meet — i.e., at the point (a,L), the function is undefined.

When looking at this equation on a graph, it will approach this coordinate but never exactly match the specified point of discontinuity.

Vizualizing Epsilon and Delta

The way to visualize epsilon and delta would be to imagine any point which lies on the function — let’s say (x1,y1) — and then the distance from a to x1 will be delta, and the distance from L to y1 will be epsilon. 

We can see this relationship as x1 + δ = a and y1 + ϵ = L

Now, you can move this point closer to the (a,L), and you will notice epsilon and delta will shrink as the x1 and y1 values get bigger, and as ϵ and δ are inversely proportional to x1 and y1

Once delta and epsilon are sufficiently small, you can determine the point of discontinuity based on the value of x1 and y1.

An easier way to understand epsilon-delta in a real-life example is to think of it as a cupcake. You can cut a cupcake in half, then cut the half into a quarter, then the quarter into an eighth, and so on.

While your cupcake section will keep getting smaller, it will never disappear.

In the end, you’ll be left with the atomic molecules of a cupcake and if you cut further, the possible creation of a black hole. But, the cupcake will never ultimately reach a zero amount. That is the easiest and yummiest way to explain epsilon-delta.

If you feel that you are terrible at math, you probably lament this fact: math is full of symbols. One common symbol that may confuse you is the big e, written as Σ. What is this symbol? What does it mean? Does it have any connection to epsilon? Let’s take a look.

Big E in Math

The Σ symbol in math is another Greek letter called sigma. While it looks similar to epsilon (ε), it has a completely different form and purpose. Epsilon is the equivalent of E in Greek, but sigma is the uppercase version of the letter S.

Mathematicians pronounce sigma as “sum,” and it means to “sum things up.” It is different from its English equivalent of summing up ideas, but it is the result of any equation.

When you use it in an equation, sigma sums up everything that appears after the symbol. Most simply, you can use it to add up equations, and it indicates that you should add the letters and numbers around it together (source).

The Other Big e in Math

Epsilon and sigma are not the only E’s in mathematics. While they might look similar, another “e” has a very different function in the field. Known as Euler’s number, this eponymous symbol is almost the opposite of epsilon.

Euler’s number is an irrational number, and it is also the base for natural logarithms. Unlike epsilon, mathematicians can define it, and they express “e” as 2.718281828459045…which continues without end.

Euler’s number is about growth. You will typically find the “e” in mathematical formulas that refer to nonlinear growth (source). The easiest way to express “e” is to understand the equation that creates it.

The “e” is equal to an unending number of factorials. For example, we would express a factorial of 5 as 1 x 2 x 3 x 4 x 5 = 120, or you would write it as 5! in an equation. It would look like this: “e” = 1/0! + 1/1! + 1/2! + 1/3! + and so on. Hence, its irrationality.

Irrational Numbers 

Irrational numbers are numbers that are not rational. They are not whole or real numbers, though they can be positive numbers. You can think of irrational numbers as numbers that we cannot express in fraction form with integer values (source).

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We can also write Irrational numbers without an end when expressed in decimal form. Pi (𝜋) is the most famous irrational number in the world, but Euler’s number is a close second.

Final Thoughts

Math is not everyone’s cup of tea, and more complicated symbols like epsilon can trip up any student. However, like all sections of mathematics, there is always light at the end of the tunnel of confusion.

Whether you have to study hard to pass mathematics or you have a natural talent for numbers, everyone has strengths and weaknesses in the academic arena.

So many are uncertain where they stand on the continuum, so a look at “Finding and Understanding Your Academic Strengths and Weaknesses” might help.

With a long history and interesting functions, epsilon is more than a backward 3. You can easily break a lot of difficult mathematical concepts down, and once you understand the foundations, even ideas like epsilon-delta begin to make sense.