Algebraic Numbers and Transcendental Numbers

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  • 0:01 Algebraic Numbers
  • 2:07 Transcendental Numbers
  • 3:49 How Common Are These Numbers?
  • 5:23 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Math is all about numbers. In this lesson, you'll learn about the two different categories of numbers, called algebraic and transcendental. You'll learn the definition of each type and find out how common each of them is.

Algebraic Numbers

In algebra, numbers fall into one of two categories: algebraic or transcendental. It's important to understand the difference between algebraic and transcendental numbers because these numbers are the basis for algebra and higher math.

Before we get into the definitions of these two categories, let's briefly review polynomials, coefficients and rational numbers. Rational numbers are numbers that can be written as the division of two integers. Recall that polynomials with rational coefficients are functions, such as x + 4 and x^2 + 3x - 1. Our coefficients - 1, 4 in the first function and 1, 3, -1 in the second function - are all rational numbers. To find the solution to a polynomial, we set the polynomial equal to 0, and then solve for the variable. An algebraic number is any number that is the solution to a polynomial with rational coefficients.

For example, 5 is an algebraic number because it is the solution to x - 5 = 0. The square root of 5 is also an algebraic number because it is the solution to x^2 - 5 = 0. The imaginary number i is also an algebraic number because it is the solution to x^2 + 1 = 0. As we can see, every algebraic number is defined by a polynomial, which is why we say that algebraic numbers are definable.

While all rational numbers are algebraic, not all algebraic numbers are rational. Algebraic numbers can be radicals, irrational numbers and even the imaginary number. As long as the number is the solution to a polynomial with rational coefficients, it is included in the category of algebraic numbers.

Transcendental Numbers

If a number is not an algebraic number, then it is considered a transcendental number. So, transcendental numbers are those numbers that are not the solutions to polynomials with rational coefficients.

Here are a few examples of transcendental numbers.

The first transcendental number that was proved was the Liouville constant, 0.11000100000000000000000100......, where there is a 1 at every n! places after the decimal point. 1! equals 1, so there is a 1 in the first place after the decimal. 2! equals 2, so there is a 1 in the second place after the decimal. 3! equals 6, so there is a 1 in the sixth place after the decimal.

The next number to be proved a transcendental number was the constant e. It was proved in the year 1873. In 1884, the number pi was proved to be a transcendental number, as well.

The Liouville constant, the constant e and the number pi are all numbers that you can't get by solving a polynomial with rational coefficients. It doesn't matter how hard you try!

Another aspect of transcendental numbers is that they are also irrational. An irrational number is a number that can't be written as the fraction of two integers. These are decimal numbers that keep going to infinity without repeating. But just because all transcendental numbers are irrational doesn't mean that all irrational numbers are transcendental.

How Common Are These Numbers?

So which type of number is more common? Both algebraic and transcendental numbers are infinite in quantity, but transcendental numbers are far more common. To help us understand why, let's use an analogy.

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