There are countably infinite algebraic numbers.
Understanding Algebraic Numbers
Algebraic numbers are numbers that are roots of a non-zero polynomial with integer coefficients. This includes familiar numbers like integers, rational numbers (fractions), and certain irrational numbers. Here's a quick breakdown:
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Definition: A complex number x is considered algebraic if it is a solution to a polynomial equation of the form:
- anxn + an-1xn-1 + ... + a1x + a0 = 0
- Where an, an-1,..., a0 are integers and an ≠ 0.
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Examples:
- All integers (e.g., 5 is a root of x - 5 = 0)
- All rational numbers (e.g., 3/2 is a root of 2x - 3 = 0)
- √2 (root of x2 - 2 = 0)
- The imaginary unit i (root of x2 + 1 = 0)
Countably Infinite: What Does it Mean?
When we say the number of algebraic numbers is "countably infinite," we mean the following:
- It is an infinite set.
- It can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3...). Essentially, you can list them out, even if that list is infinite.
- This contrasts with "uncountably infinite" sets, like the real numbers or complex numbers, which are so large that you cannot create such a list.
Countability of Algebraic Numbers: Key Insight
The set of algebraic numbers, although infinite, is countable. This is because:
- The set of polynomials with integer coefficients is countable.
- Each polynomial has only a finite number of roots.
- The union of countably many finite sets (roots of polynomials) results in a countable set.
The Contrast with Complex Numbers
It is critical to understand that the set of complex numbers is uncountably infinite. The set of algebraic numbers, while infinite, is just a tiny fraction of all complex numbers. The reference states the set of algebraic numbers has measure zero, which means it is negligibly small when compared to the set of all complex numbers under the Lebesgue measure. This underscores the fact that most complex numbers are not algebraic.
Practical Implications
While the number of algebraic numbers is infinite, many familiar numbers are algebraic. However, the overwhelming majority of numbers (like π and e ) are not algebraic; they are called transcendental numbers. This shows that while algebraic numbers are numerous, they're also relatively "special" within the vast space of complex numbers.
| Feature | Algebraic Numbers | Complex Numbers |
|---|---|---|
| Countability | Countably Infinite | Uncountably Infinite |
| Size | Small subset of complex numbers | Largest set |
| Measure | Zero | Non-zero |
| Examples | Integers, rational numbers, √2, i | Most numbers are transcendental |
| Definition | Root of a polynomial with integer coefficients | all numbers of the form a+bi |
