How Much Mathematics Do You Need To Learn Programming

Mathematics is a subject which most students afraid of because of its complexity but once you understood what mathematics is, then you will happily learn mathematics.

In most of the schools and colleges mathematics is teached for passing exams only without telling you applications of mathematics which can development your interest.

As of now if you are fascinated with AI on how a machine can answer our question so accurately, then you will be amazed that it is possible due to mathematics.

These are some topics with their applications which you can explore more. They will give you a basic highlight on how much mathematics do you need for programming.

Mathematical topics which you have to learn for programming

From Programming i not mean web development or graphic designing, if you any of them you can leave here and enjoy programming without mathematics, Mathematics is Important for DSA(Data Structures and Algorithms), AI(Artificial Intelligence), ML(Machine Learning).

1. Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations such as:{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}

linear maps such as:{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}

and their representations in vector spaces and through matrices.^[1][2][3]

Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to function spaces.

2. Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point.

Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. The indefinite integral, also known as the antiderivative, is the inverse operation to the derivative.

3. Conditional Probability

In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. 

This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B.

If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B“, or “the probability of A under the condition B“, is usually written as P(A|B) or occasionally P_B(A).

This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the “given” one happening (how many times A occurs rather than not assuming B has occurred):

.

For example,

the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing.

For example,

the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that P(Cough) = 5% and P(Cough|Sick) = 75 %. Although there is a relationship between A and B in this example, such a relationship or dependence between A and B is not necessary, nor do they have to occur simultaneously.

P(A|B) may or may not be equal to P(A), i.e., the unconditional probability or absolute probability of A. If P(A|B) = P(A),

then events A and B are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. P(A|B) (the conditional probability of A given B) typically differs from P(B|A).

For example,

if a person has dengue fever, the person might have a 90% chance of being tested as positive for the disease. In this case, what is being measured is that if event B (having dengue) has occurred, the probability of A (tested as positive) given that B occurred is 90%, simply writing P(A|B) = 90%. Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high false positive rates. In this case, the probability of the event B (having dengue) given that the event A (testing positive) has occurred is 15% or P(B|A) = 15%.

It should be apparent now that falsely equating the two probabilities can lead to various errors of reasoning, which is commonly seen through base rate fallacies.

While conditional probabilities can provide extremely useful information, limited information is often supplied or at hand. Therefore, it can be useful to reverse or convert a conditional probability using Bayes’ theorem: 

Another option is to display conditional probabilities in a conditional probability table to illuminate the relationship between events.

Linear Algebra Applications:

1. Computer Vision & Deep Learning
   • Image recognition (face detection, object tracking) using matrices.
   • Convolutional Neural Networks (CNNs) rely on matrix multiplications.
2. Quantum Computing
   • Quantum states represented as vectors; operations use matrices.
3. Google PageRank Algorithm
   • Uses eigenvectors to rank web pages based on links.

Differentiation (Derivatives) Applications:

1. Rocket Launch Trajectories
   • Calculating acceleration from velocity changes in space travel.
2. Economics & Cost Optimization
   • Finding max profit/min cost by optimizing revenue and expenses.
3. AI Training (Gradient Descent)
   • Backpropagation in neural networks adjusts weights using derivatives.

Integration Applications:

1. Autonomous Drones & Robots
   • Calculating precise movements based on acceleration and position data.
2. MRI Scans & Image Processing
   • Reconstructing 3D images from multiple 2D slices using integrals.
3. Fluid Dynamics & Aerodynamics
   • Computing lift force in airplanes by integrating pressure differences.

Conditional Probability Applications:

1. AI Chatbots & Language Models
   • Predicting the next word in a sentence based on previous words (Markov Models).
2. Fraud Detection in Banking
   • Estimating fraud risk based on past transaction patterns.
3. Genetics & DNA Analysis
   • Probability of inheriting diseases given family medical history.

These are some topics which can help you develop interest in mathematics and also help in developing programming skills, Thank you for reading my blog, if you liked it, share it and give us a beautiful comment of feedback.