What Does the Funny Looking E in Math Mean
With so many signs and symbols in mathematics, information technology is sometimes hard to decipher or fifty-fifty understand what each ane stands for. For instance, the backward 3 symbol (ε) — what does information technology hateful, and how do mathematicians use it in equations?
The ε symbol, too known as epsilon, represents the closest number to zero, however it is non zilch. It is not a constant number, and it is variable depending on the equation. You will detect it in many fields of mathematics but nearly commonly in calculus and algebra. While the ε is a variable, due to its changing nature, information technology is challenging for anyone to define.
Welcome to the globe of mathematics, where signs and symbols dominate a pupil'south examination paper and take over a scientist or mathematician'southward work journal. For those of y'all who thrive in the world of numbers, let'due south embark on this journeying of agreement 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 modernistic society, and the epsilon symbol is one of them (source).
Epsilon is function of the Phoenician alphabet that the Greeks co-opted from the original symbol when they created their own alphabet. Information technology still resembles the original Phoenician letter today when written in the lower-case grade.
In its upper-case form, it looks like a large E. Similar the English alphabet, it is the 5th letter of the Greek alphabet and information technology creates a similar phonetic sound. As it is the fifth letter of the alphabet, many associate it with the number 5.
Initially, they called it "ei," naming information technology after the sound that it made. In the Eye Ages, European scholars named information technology "epsilon" to distinguish it from similar-sounding letters.
With such a long history, information technology is understandable that epsilon plays a major function in mathematics, astrology, and science, among other fields.
How to Utilise Epsilon
Paul Erdös, a mathematician, may have used the all-time illustration to depict epsilon's function — he referred to epsilon as his "children," and the metaphor is helpful to those of us who are non gifted in math.
When y'all think of children, you may be thinking of delightful little bundles of energy running effectually a garden or, mayhap, some demon spawn that throws things at your canis familiaris from over the fence.
Just like children, the epsilon never remains the same for long — and y'all volition likely become a headache trying to sympathise both children and the epsilon — the larger they grow, the further abroad from y'all they get.
As mentioned previously, epsilon is a small number, almost equally small as naught, but never achieving pure nothingness like zero.
For those long-suffering algebra students, epsilon is a variable like to Ten 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 nigh the real numbers (again, just plain numbers) we utilise.
The idea of real numbers is an umbrella term and tin can embrace a variety of number types. By definition, real numbers are whatever 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 instance of a real number. Since epsilon does not have a set value but could be a range of numbers, you tin can classify it equally a real number.
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 lot can quantify everything countable with positive numbers. Y'all also utilise positive numbers to measure space, volume, distance, and even force and energy. Substantially, anything we can quantify as a number above zippo is a positive number.
Since epsilon refers to a number greater than goose egg but which is not zippo, information technology is a perfect example of a multitude of positive numbers.
If yous detect yourself confused with how to reference the number nada, check out "Zeroes or Zeros: Understanding the Substantive, Verb, and Adjective" to know which one to use when you cannot employ the number itself.
What Is the Value of Epsilon?
Every bit mentioned previously, epsilon can refer to any range of numbers that are non nil, but its nigh full general usage is every bit a stand-in for a number between 0 and 1.
Instead of using the specific fraction or decimal to express an verbal number, you lot 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 null, this tin can be an infinite amount of numbers.
Epsilon can exist 0.one 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 hard or cumbersome to write.
Why Exercise We Need Epsilon?
Epsilon is the Plato, Aristotle, and Socrates of mathematics. You will not frequently notice 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 nigh common usage of epsilon is the more easily understood stand-in for a small number, merely that is non its just purpose.
Within mathematics, there is the concept of limits. A limit is likewise 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. Information technology requires some breaking downwards to sympathise, so let'due south first past taking a expect at the graph below.
Functions
Within mathematics, a function refers to a procedure in which there is an input, a relationship that takes place, and an output.
Functions crave x and y every bit function of the equation, and they are usually expressed on a Cartesian plane (fatigued in black above).
In an equation, you will mostly present functions as f. While x usually stands solitary in an equation, y'all can express y every bit f(x), which is drawn in red in a higher place.
For example, x multiplied past two is an example of a part. Ten is the input, x 2 is the human 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, so the output is 538.
Functions are not only restricted to multiplication simply rather all forms of mathematical operations such every bit addition, subtraction, sectionalisation, and so on.
Limits
Limits in mathematics means that you are approaching a specific 10 coordinate on a function line. This means that something is approaching along the f(ten) line simply never hitting the x coordinate, shown in the limit.
Commonly, when evaluating the coordinates of a specific betoken on f(10), you lot would substitute the ten in the role with a value and determine what the f(ten) output would be, but you cannot practise this when in that location is a discontinuity or break in the graph.
In that case, you would demand to reference limits to determine where the discontinuity is.
Epsilon is an example of a limit because it approaches null only never becomes small-scale enough to be nix. When y'all nowadays epsilon-delta as an equation, you would write this as:
lim f(x) = L
x￫a
F(ten) is a function of x, a is the x-value of the bespeak being approached, and L is the y-value existence approached.
The graph above is discontinuous at the coordinate where a and L would meet — i.e., at the signal (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 exist to imagine any point which lies on the function — allow's say (x1,y1) — and so the altitude from a to x1 will be delta, and the altitude from L to y1 volition be epsilon.
We can see this human relationship every bit x1 + δ = a and y1 + ϵ = L
Now, you can movement this indicate closer to the (a,L), and y'all will discover epsilon and delta volition compress 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 lot tin determine the betoken of discontinuity based on the value of x1 and y1.
An easier way to sympathize epsilon-delta in a existent-life example is to think of it as a cupcake. You tin cut a cupcake in one-half, and then cutting the half into a quarter, so the quarter into an 8th, then on.
While your cupcake department will continue getting smaller, it will never disappear.
In the finish, 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 corporeality. That is the easiest and yummiest way to explain epsilon-delta.
If you feel that yous are terrible at math, you probably complaining this fact: math is full of symbols. One common symbol that may confuse you is the big e, written every bit Σ. What is this symbol? What does it mean? Does it have any connection to epsilon? Let's take a wait.
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 class and purpose. Epsilon is the equivalent of E in Greek, only sigma is the majuscule version of the letter Due south.
Mathematicians pronounce sigma as "sum," and it means to "sum things up." It is dissimilar from its English equivalent of summing up ideas, simply it is the result of any equation.
When you utilise it in an equation, sigma sums up everything that appears after the symbol. Most just, you tin employ it to add together up equations, and it indicates that you lot should add the letters and numbers around it together (source).
The Other Large east in Math
Epsilon and sigma are not the only E'southward in mathematics. While they might wait similar, some other "e" has a very different function in the field. Known as Euler's number, this eponymous symbol is about 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 limited "e" as two.718281828459045…which continues without end.
Euler's number is nigh growth. You volition typically find the "east" in mathematical formulas that refer to nonlinear growth (source). The easiest way to limited "e" is to sympathise the equation that creates information technology.
The "east" is equal to an unending number of factorials. For example, we would express a factorial of five equally ane x 2 x 3 x 4 x v = 120, or you would write it every bit 5! in an equation. Information technology would look like this: "e" = 1/0! + i/1! + 1/2! + 1/3! + and then on. Hence, its irrationality.
Irrational Numbers
Irrational numbers are numbers that are not rational. They are not whole or real numbers, though they can exist positive numbers. You can remember of irrational numbers as numbers that we cannot express in fraction form with integer values (source).
This article was written for strategiesforparents.com.
We tin besides write Irrational numbers without an cease when expressed in decimal form. Pi (𝜋) is the nearly famous irrational number in the world, but Euler'south number is a close second.
Final Thoughts
Math is not everyone'due south cup of tea, and more complicated symbols like epsilon tin trip upwards any student. Nonetheless, like all sections of mathematics, there is always lite at the stop of the tunnel of confusion.
Whether you accept to study difficult 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 three. You can hands interruption a lot of difficult mathematical concepts down, and once you sympathise the foundations, even ideas similar epsilon-delta begin to make sense.
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