ELI5: Difference Between Algebra and Calculus

Algebra and Calculus both belong to different branches of mathematics but are closely related to each other. Applying basic algebraic formulas and equations, we can find solutions to many of our day-to-day problems. 

Calculus is mostly applied in professional fields due to its capacity for finding solutions to many types of structures. Algebraic formulas and concepts are used in solving almost all mathematical problems including arithmetic, trigonometry, and Calculus, apart from their use in finding solutions to problems of other branches of science and technology. 

Although these branches of math are indispensable, they can get very intimidating for beginners. Alternative methods of teaching like games and activities might help to make it more interesting for the learner. 

Algebra vs Calculus: Basic difference

Although Algebra and Calculus are interrelated, there is a definite difference between the two.

Calculus deals with operations on functions and their derivatives, whereas algebra deals with operations on variables and numbers. Algebra assists in finding the slope of a line, while calculus is useful for finding the slope of a curve. Algebra helps in finding the direction of motion along a straight line, whereas calculus does the same along any curve.

Algebra is based on the goal of operations on equations, often with the intent of solving for unknown values. Calculus is where we begin to quantify complex relationships. Calculus is all based on change. We mainly use limits, derivates, and integrals. Limits simply ask what value y approaches as x approaches some value. Derivates are the way that we find a function’s rate of change at a given moment. An integral is a mathematical object that can be interpreted as an area or a generalization of an area. 

Difficulties a student might face while studying Algebra

Some students might find it difficult to understand the concepts of algebra. Like, finding it difficult to solve the problems or finding it hard to remember the formulas and equations. While the reasons could be many, the solution is one- Practice!

A study identified the difficulties that students faced when solving algebraic problems, 

  • visual-spatial skill
  • text difficulties and unfamiliar context
  • language skill
  • recognition difficulties
  • sign substitution error and 
  • translation of words or diagrams to algebra and vice versa

All these various difficulties faced while studying algebra could be because of 

1. The nonvisual nature of algebra

Algebra is the branch of mathematics that uses symbols and variables to represent numbers and quantities in equations and formulas. It is considered one of the most difficult subjects due to its nonvisual concepts. Children are taught Math with very visual concepts till the elementary stage. Hence the nonvisual concept of algebra seems difficult.

2. The arbitrary nature of algebra

Algebra can be quite confusing for students as it often contains a lot of rules that seem arbitrary. For example, the rule that x + y = y + x is not always true in algebra. This can be quite confusing for students who are used to the commutative property of addition from arithmetic.

3. The Relationship Between Symbolism, System, and Question

Algebra is a symbolic language that represents relationships between quantities. These relationships can be quite complex, and it can be difficult for students to understand how the symbols relate to the quantities they represent.

4. The Complex Nature of Algebra

Algebra is a very abstract subject, and it can be difficult for students to grasp the concepts without a solid understanding of the underlying principles. This can make algebra very difficult for students struggling with other mathematics subjects.

Difficulties a student might face while studying Calculus

Calculus seems difficult because it requires a lot of hard work. Moreover, it is also challenging if the students have never been exposed to them before. The computations can be very tedious and time-consuming. And it is easy to make mistakes when working with calculus, which can lead to incorrect results. Some of these challenges could be:

  • It needs mastery over algebra: Calculus is a mathematics subject that requires a strong mastery of algebra. If a student struggles with algebra, they will likely find calculus equally challenging.
  • It is more conceptual than basic math courses: It is more conceptual than basic math courses: Calculus builds on the concepts learned in lower-level math courses but is much more abstract. This can make it difficult for students to grasp the material.
  • Has several highly abstract ideas: Calculus involves a lot of theoretical concepts that can be difficult to understand. Additionally, there are many formulas and equations that students need to memorize.
  • It is not always intuitive and requires strong background basic information: Unlike some other subjects, calculus is not always easy to visualize or understand without further explanation. This can make it tough for students to learn on their own.
  • Lack of understanding of the nature of the subject: One of the biggest challenges for students is simply not understanding what calculus is and how it works. This can make the material seem insurmountable.
  • An individual with dyscalculia might find it difficult to understand calculus: Dyscalculia is a learning disorder that affects an individual’s ability to do math. Those with dyscalculia might find it especially hard to understand the abstract concepts involved in calculus.

What is the Difference Between Calculus and Algebra?

MeaningCalculus is the study of change and movementAlgebra is the study of equations and variables
NatureCalculus is a dynamic disciplineAlgebra is a static discipline
ApproachCalculus is about finding relationshipsAlgebra is about finding solutions
CauseCalculus is focused on understanding how things changeAlgebra is focused on solving equations

Is algebra used in calculus?

Yes, a thorough knowledge of basic algebra is required for calculus. 

Calculus is algebra and trigonometry with limits. There is often only one step in the problem that actually involves calculus, the rest is simplifying using algebra and trigonometry.

One needs to be able to factor expressions and equations easily. Many of the problems in calculus will involve finding the roots of a function and for the most part, that means factoring. 

Linear and non-linear functions are central to calculus and machine learning, and many calculus problems involve their use. Fundamental concepts from algebra and trigonometry provide the foundations for calculus and will become especially important as we tackle more advanced calculus topics.      

Application of algebra vs application of calculus

Differential Calculus allows one to predict the motion of bodies such as stars, rockets, planes, trains, cars, boats, and particles. We couldn’t navigate a ship or a car without it.

Integral Calculus is used to calculate the quantity where the rate of change is known. For example, we can calculate the predicted population of a country if we know the birth rate even if it changes over time. We can calculate how much fuel a rocket will need to get to the moon and a car to get to the petrol station if we know its velocity and acceleration over the time required.

Algebra is the building block on which most mathematics is built. It uses letters and symbols to replace actual numbers in formulae to make them applicable across multiple scenarios. For example, by using v= u + at we can calculate the predicted velocity of any car at any time if we know its starting velocity, its acceleration, and the time it has been accelerating for.

Summing up,

We can now conclude that no matter how complicated these subjects may seem, algebra and calculus form a very important part of our daily life. Sound knowledge of algebra is required for calculus. Algebra is an important life skill worth understanding well. It moves us beyond basic math and prepares us for statistics and calculus. It is useful for many jobs as it is useful in analyzing information. It also reinforces logical thinking, and this serves as the basis for calculus that helps us to understand other concepts in subjects like physics.

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