What is algebra? When you equate two expressions, they form an equation. Algebra studies the basic properties of numbers and equations and introduces the rules that we must follow to solve these equations. Algebra is the branch of mathematics that helps us to form mathematical expressions in a meaningful way.

To “solve an equation” means to determine what value(s) each unknown variable must hold to ensure that both sides of the equation are equal. Algebra is fundamentally concerned with manipulating equations to solve for unknown variables.

Unknown variables are the letters and symbols that take the place of numbers in equations. For example, three common examples of variables are $x$, $y$, and $z$.

Algebra teaches us how to manipulate variables using operations such as:

Addition

Subtraction

Multiplication

Division

At its core, algebra involves understanding what operations can be used to manipulate a set of elements. A 9th-century Muslim mathematician named Muhammad ibn Musa al-Khwarizmin invented Algebra.

Some examples of key milestones in learning algebra are:

Forming algebraic expressions and equations using a combination of constants, variables, and operators

Defining and graphing functions

Using graphs to solve equations

Learning the algebraic properties for real numbers:

- The Commutative Law of Addition

- The Commutative Law of Multiplication

- The Associative Law of Addition

- The Associative Law of Multiplication

- The Distributive Law of Multiplication

- The Additive Identity

- The Multiplicative Identity

- The Additive Inverse

- The Reciprocal Rule

Here’s more about real numbers:

Understanding the differences between linear, cubic, and quadratic equations

Manipulating exponents and logarithms in an equation

Factoring polynomials

Solving systems of linear equations and inequalities, and using the properties of matrices

What Is Calculus?

What is calculus? Calculus is the study of rates of change. Rates of change are used to describe the change that occurs in one variable as another variable changes. Calculus is specifically concerned with instantaneous rates of change. The instantaneous rate of change of a function is called its derivative. In differential calculus, the process of calculating a derivative is called differentiation.

We can conceptualize derivatives in many ways. For example, we can think about the derivative as the instantaneous rate of change of a function, the slope of the tangent line at a point, the limit of the difference quotient as the change in $x$ approaches zero, or even velocity.

Integral calculus is another primary subject of study in calculus. You can think about integration as the reverse operation of differentiation.

There are two types of integrals: indefinite and definite.

Indefinite integrals are used to find the antiderivative function, while definite integrals are used to calculate the area below the curve over a specific interval.

The Fundamental Theorem of Calculus is the unifying thread that links differential and integral calculus. We celebrate this theorem as one of the most pivotal discoveries of modern mathematics. Isaac Newton and Gottfried Leibniz invented calculus in the late 17th century.

Dr. Hannah Fry explains more about the theorem:

Some examples of key milestones in learning calculus are:

Calculating the volume of a solid of revolution using the disc method, washer method, or shell method

Main Differences Between Algebra and Calculus

What is the difference between calculus and algebra? Here are three fundamental differences between these different branches of mathematics.

Algebra is primarily concerned with solving equations, while calculus is primarily concerned with calculating the instantaneous rate of change of functions. For example, algebra allows us to calculate the slope of a straight line, which is called the average rate of change. Calculus allows us to calculate the slope of a curve at a point, which is called the instantaneous rate of change.

The study of calculus provides the theory for how we can solve a problem, while the study of algebra provides the operational tools needed to reach the final solution. For example, the study of calculus has provided us with the basic differentiation formulas for certain types of functions. These formulas are We define and apply these formulas by using algebraic operations.

It’s been more than 4,000 years since the invention of algebra, while the invention of calculus is relatively young, at less than 400 years since its discovery. As a result, people often consider algebra to be more approachable and easier to understand than calculus.

Similarities Between Algebra and Calculus

How are algebra and calculus similar? Here are two similarities between these branches of mathematics:

Both algebra and calculus help us to represent real-life scenarios in the form of equations. Both branches of mathematics help us to translate human problems into mathematical statements, which have the potential to create meaningful impacts in our lives.

The rules, properties, and operations of algebra are also essential tools in calculus. For example, algebra helps us to solve limits and simplify integrals in calculus. Before learning calculus, you should be able to manipulate algebraic expressions, define functions, and use basic trigonometry. Algebra is an essential prerequisite to learning calculus.

Everyday Examples of Algebra and Calculus

We use algebra and calculus every day in a broad range of human activities. Some examples include grocery shopping, cooking, construction, engineering, computer science, and physics. Here are two mathematical problems that exemplify the everyday use of algebra and calculus.

Example 1 - Algebra in Everyday Life

You are at the movies, and you have $17 dollars to buy snacks for you and your friends. Each bag of popcorn costs $3.50 and each bar of candy costs $1.25. You want to buy 2 bags of popcorn to share. How many bars of candy can you buy for your friends with the money left over?

Solution

We can build an equation to represent this word problem. Our equation looks like this:

$3.50x + 1.25y = 17$

Since you already know how many bags of popcorn you want to buy, we can substitute $x = 2$ into the equation:

$3.50 \cdot 2 + 1.25y = 17$

$7 + 1.25y = 17$

Now, we can solve for $y$. First, we’ll subtract 7 from both sides of the equation. Then, we’ll divide by 1.25.

$1.25y = 10$

$y = 8$

Using algebra, we’ve determined that you can buy 8 candy bars for your friends if you buy 2 bags of popcorn. Let’s check our work by plugging $x = 2$ and $y = 8$ into the equation. We need to confirm that both sides of the equation are the same.

$3.50x + 1.25y = 17$

$3.50 \cdot 2 + 1.25 \cdot 8 = 17$

$7 + 10 = 17$

$17 = 17$

Example 2 - Calculus in Everyday Life

An epidemiologist in a small town models the spread of a severe flu by using reported case numbers, recovery rates, and the current number of susceptible, infected, and immune individuals. The epidemiologist proposes that the number of people in the town infected by the flu, $t$ days after the flu began to spread, can be represented by this function:

$p(t) = t^3 + 6t$

The town’s hospital wants to prepare as best they can for an influx of patients and asks the epidemiologist to provide estimations for the daily infection rates. Using the epidemiologist’s model, how many people should the hospital expect to be newly infected with the flu at the beginning of day 5?

To determine the number of people newly infected with the flu at the beginning of day 5, we must determine the instantaneous rate of change of people infected with the flu at $t = 5$. This question is asking for the instantaneous rate of change of $p(t)$ at $t = 5$.

To find the instantaneous rate of change of $p(t)$ at $t = 5$, we need to take the derivative of $p(t)$. To find $p’(t)$, the general derivative function, we’ll need to use the power rule for derivatives, which states that $\frac{d}{dx}(x^n) = nx^{n-1}$.

So, our derivative function is:

$p’(t) = 3t^{3-1} + 1 \cdot 6t^{1-1}$

$p’(t) = 3t^2 + 6t^0$

$p’(t) = 3t^2 + 6$

To find the instantaneous rate of change at $t = 5$, we plug $t = 5$ into $p’(t)$.

$p’(5) = 3(5)^2 + 6$

$p’(5) = 3 \cdot 25 + 6$

$p’(5) = 75 + 6$

$p’(5) = 81$

So, using the epidemiologist’s model approximation, the flu will newly infect 81 people at the beginning of day 5.

Frequently Asked Questions

Should I learn algebra first or calculus?

It is essential to learn algebra before learning calculus. We execute many mathematical concepts in calculus through the algebraic manipulation of functions. Often, we learn the algebraic concepts required in calculus in a precalculus course. Most high school calculus courses require algebra or precalculus as a prerequisite to enrolling in the class.

Is algebra harder than calculus?

We often consider calculus to be more difficult than algebra. Algebra courses explore the many operations, properties, and rules that can be used to manipulate equations. Calculus courses apply algebraic operations to functions in a more complex way.

Does calculus involve algebra?

Yes, calculus involves quite a bit of algebra. Before learning calculus, math students should be familiar with exponential, logarithmic, and trigonometric functions, and know how to solve linear, cubic, and quadratic equations. These concepts aid in calculating derivatives, evaluating limits, solving integrals, and much more! In multivariable calculus, linear algebra is especially useful.