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Real and Complex Numbers
Real numbers are the real-world numbers that we usually deal with excluding units like degrees, inches, etc. that go with them. These are numbers that can be found on the number line that extends to infinity. Real numbers include both negative and positive values. Complex numbers on the other hand include i. Examples of complex numbers are 2i, 3.4.5i, -ni, etc. Real numbers also fall under complex numbers. A real number n can be represented as n + 0i, which is a complex number. Complex numbers are vital in algebra and the computation of polynomial equations.
Metric Spaces
A metric space is an abstract set that has a distance function. This distance function is known as a metric. The following properties should hold when a metric specifies the distance (non-negative) between any two points:
- If the two points are similar, then the distance between them is zero
- The distance from point one to point two is the same as the distance from point two to point one.
- Triangle inequality – The total distance from point one to point two and from point two to point three is equal to or greater than the distance from point one to point three.
Limits and Convergence
A limit is a value that a sequence attains when the number of terms is infinity. However, this behavior is not exhibited by all sequences. Sequences that have this attribute are known are convergent sequences while those that do not show this behavior are known as divergent. Since limits highlight the long-term characteristics of a sequence, they are quite useful in unbounding them.