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Jul 10, 2024 | Almaden

Welcome to our beginner-friendly guide to the delta symbol (Δ) in math! Learn about its background, how it’s used in various math subjects, and take a short quiz to test your knowledge.

Delta, represented by symbols Δ (uppercase delta) and δ (lowercase delta), is a fundamental math concept that students meet at various grade levels and in different branches of mathematics.

It helps us show changes or differences in relationships between numbers or shapes, and we can use it in algebra, geometry, calculus, and beyond.

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The delta symbol first appeared in ancient times and has since then traveled through history and across continents. Originating from the Phoenician letter “daleth” or “delt” meaning “door,” the delta symbol (Δ, δ) found its place in the ancient Greek alphabet as the fourth letter.

While the earliest usage of the delta symbol in mathematics is uncertain, it became standardized during the 18th and 19th centuries. Prolific mathematicians of their time, such as Leonhard Euler and Augustin-Louis Cauchy, used delta in their works on calculus and most likely set the stage for its adoption across various branches of mathematics.

In math, both uppercase (Δ) and lowercase delta (δ) mostly represent change or difference. The versatility of this symbol allows mathematicians to use it as a marker for variations in quantities, functions, or geometric properties.

Here are several common applications of the delta symbol across various math branches:

In algebra, the discriminant, represented as uppercase delta (Δ), is **a value calculated from the coefficients of a quadratic equation**. It is used to determine the nature of the solutions to the equation.

- If Δ is greater than zero, the equation has two distinct real roots
- If Δ equals zero, it has one real root
- If Δ is less than zero, it has no real roots

The discriminant provides a quick way to understand whether a quadratic equation has solutions and, if so, how many, without having to graph the equation.

In algebra, a quadratic equation has the general form:

*ax ^{2 }+ bx + c = 0*

Here, *a*
and *b*
are coefficients, *c*
is a constant, *x*
is a variable.

The discriminant, denoted by Δ, is calculated using the formula:

*Δ = b ^{2} - 4ac*

Let’s say we have the quadratic equation *2x ^{2} + 3x - 2 = 0*

Now, we can calculate the discriminant:

*Δ = (3) ^{2} - 4(2)(-2)*

*= 9 + 16*

*= 25*

Since the discriminant is greater than zero (Δ > 0), this means the quadratic equation has two distinct real roots.

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In calculus, the delta symbol takes on another important role, particularly in relation to partial derivatives.

**The derivative of a function **measures the smallest, near-zero changes in one variable, represented by the letter “d.”

Partial derivatives, on the other hand, focus on how a function changes in relation to specific variables while keeping other variables constant. In this case, we use the lowercase delta notation (δ) to indicate the partial derivative.

Let’s say we have a function* f (x, y) = x ^{2} + y.*

We want to see *f* how
changes with
while keeping *y*
constant.

That’s where the delta symbol comes in. So, δf/δx (read as “delta f over delta x”) tells us exactly what we want to know; it’s a way to focus on how changes with without considering what’s happening with at that moment.

Here’s a simple word problem to illustrate how it works:

Let’s say you’re calculating the total cost of a picnic basket based on two factors: the number of sandwiches and the price of lemonade .

Your cost formula is:

*C (n, p) = 2n + 3p*

Here, *n*
represents the number of sandwiches, and *p*
represents the price of lemonade per bottle.

**Question: **

How does the total cost change if the number of sandwiches increases while the price of lemonade remains constant?

**Solution: **

To find out how the total cost changes with respect to the number of sandwiches *(n)*
, while keeping the price of lemonade *(p)*
constant, we calculate the partial derivative *δC/δn*.

Given *C (n, p) = 2n + 3p*, to find δC/δn, we differentiate *C* with respect to *n* while treating *p* as a constant:

*δC/δn = 2*

This means that the rate of change of the total cost with respect to the number of sandwiches is constant and equal to 2. So, for every additional sandwich , the total cost increases by $2, while keeping the price of lemonade constant.

In advanced linear-algebra concepts, the Kronecker delta symbol, denoted by δij, indicates a correlation between two integral variables.

δij equals 1 when the two variables are equal and 0 when they are not equal. It is a valuable tool for expressing relationships between indices or variables in matrices, tensors, and other discrete systems.

Since the Kronecker delta involves advanced math, we’ll keep it at this for now.

Since the symbol’s appearance may sometimes seem misleading, it’s important to learn how to distinguish between delta and similar-looking math symbols. Here are a few that are commonly confused with delta:

**Nabla (∇)**

In vector calculus, the nabla symbol (∇), also known as “del,” looks like an inverted delta and serves as an operator. Unlike delta (∆), which represents a change or a difference, the nabla symbol (∇) performs vector operations. It is used to determine the gradient of a function, the curl and divergence of a vector field, and other properties of multidimensional functions.

**Partial Derivative Operator (∂)**

In calculus, the lowercase delta symbol δ has a counterpart known as the “partial derivative operator” and represented by the symbol “∂” which looks like a mirror-image of lowercase delta. We use the partial derivative operator to calculate the derivative of a function with multiple variables, focusing on how it changes with one variable while keeping the others constant.

Create your own user feedback surveyFind answers to common questions about the meaning, uses, and significance of the delta symbol in various mathematical contexts.

The delta symbol (∆) originated from the Greek alphabet and was derived from the Phoenician letter “daleth,” meaning “door.”

Though the exact time when the delta symbol was first used in math is not certain, it was notably used by 17th-century mathematician Gottfried Wilhelm Leibniz. The symbol became standardized in the 18th and 19th centuries, with mathematicians like Leonhard Euler and Augustin-Louis Cauchy adopting it into their work.

** **

Students usually start using delta in math class during high school, but some may learn about it earlier, depending on their school’s curriculum.

** **

Yes, delta is also used in fields like science, engineering, and finance to represent changes or differences.

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