The Most Helpful Theorems In Math That You Need To Know In 2023

Introduction

Mathematics can be a challenging subject for many students. However, understanding some of the most helpful theorems in math can make the subject much easier to grasp. In this article, we will explore some of the most important theorems that will help you succeed in math.

The Pythagorean Theorem

The Pythagorean Theorem is one of the most well-known theorems in math. It states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This theorem has a wide range of applications in fields such as geometry, physics, and engineering.

Example:

If a right-angled triangle has sides of length 3 and 4, what is the length of the hypotenuse? Using the Pythagorean Theorem, we can calculate that the length of the hypotenuse is 5 (3^2 + 4^2 = 5^2).

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that differentiation and integration are inverse operations. This theorem is essential for understanding calculus and is used extensively in fields such as physics, engineering, and economics.

Example:

If f(x) = x^2, what is the integral of f(x) with respect to x? Using the Fundamental Theorem of Calculus, we can calculate that the integral of f(x) is (1/3)x^3 + C, where C is a constant.

Fermat’s Little Theorem

Fermat’s Little Theorem is a theorem in number theory that states that if p is a prime number and a is any integer, then a^p-a is an integer multiple of p. This theorem is used in cryptography and is essential for understanding modular arithmetic.

Example:

If p = 7 and a = 3, what is the value of 3^7-3? Using Fermat’s Little Theorem, we can calculate that 3^7-3 is a multiple of 7. Therefore, 3^7-3 is equal to 0 mod 7.

The Binomial Theorem

The Binomial Theorem is a theorem in algebra that describes the expansion of powers of binomials. This theorem is used extensively in probability theory and is essential for understanding the concept of probability.

Example:

What is the expansion of (a + b)^3? Using the Binomial Theorem, we can expand (a + b)^3 as a^3 + 3a^2b + 3ab^2 + b^3.

The Law of Cosines

The Law of Cosines is a theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This theorem is used extensively in navigation and engineering.

Example:

If a triangle has sides of length 3, 4, and 5, what is the cosine of the angle opposite the side of length 5? Using the Law of Cosines, we can calculate that the cosine of the angle opposite the side of length 5 is 3/5.

Euler’s Formula

Euler’s Formula is a theorem in complex analysis that relates the exponential function to trigonometric functions. This theorem is used extensively in fields such as physics, engineering, and computer science.

Example:

What is the value of e^(iπ)? Using Euler’s Formula, we can calculate that e^(iπ) is equal to -1.

The Central Limit Theorem

The Central Limit Theorem is a theorem in probability theory that states that the sum of a large number of independent random variables will be approximately normally distributed. This theorem is used extensively in statistics and is essential for understanding the concept of sampling.

Example:

If we roll a fair six-sided die 100 times, what is the probability that the sum of the rolls will be between 300 and 400? Using the Central Limit Theorem, we can approximate the sum of the rolls as a normal distribution with a mean of 350 and a standard deviation of 17.32. Therefore, the probability that the sum of the rolls will be between 300 and 400 is approximately 0.0228.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is a theorem in algebra that states that every non-constant polynomial has at least one complex root. This theorem is essential for understanding the properties of polynomials and is used extensively in fields such as engineering and physics.

Example:

What are the roots of the polynomial x^3 – 3x^2 + 3x – 1? Using the Fundamental Theorem of Algebra, we know that this polynomial has three complex roots.

The Law of Sines

The Law of Sines is a theorem in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. This theorem is used extensively in navigation and engineering.

Example:

If a triangle has sides of length 3, 4, and 5, what is the sine of the angle opposite the side of length 5? Using the Law of Sines, we can calculate that the sine of the angle opposite the side of length 5 is 4/5.

Conclusion

In conclusion, understanding these helpful theorems in math can make the subject much easier to grasp. These theorems have a wide range of applications in fields such as physics, engineering, and computer science. By mastering these theorems, you can succeed in math and achieve your academic and career goals.