- Discussion Results: what is a number?
- Something about group theory
- This is more like a way of telling us how to use numbers, not really a good definition.
- Set up bounds to define things
- Different classes (natural, real, imaginary)
- Where do you draw the boundaries between objects?
- A way to quantify the nature of living and reality
- Something about group theory
- Number Systems
- We want them to be desirable and group-like
- Types
- Natural Numbers
- Integers greater than zero
- Whole Numbers
- Natural Numbers + 0
- 0 is the hole.
- Integers
- { …, -2, -1, 0, 1, 2, … }
- Good for algebra, we'll see later
- Rationals
- Like \(\frac{1}{2}\).
- A ratio/fraction/quotient of integers
- Real
- Like \(\pi\)
- A number on the number line
- A number that can be a distance to something.
- A good enough definition that isn't "real analysis"
- Complex Numbers
- Like \(5i\)
- There will be many complex numbers
- Matrices with complex numbers can be different from real numbers
- Complex plane
- Hamaltonian numbers music video? #curiosity
- Natural Numbers
- Why do we want more numbers?
- Why Zero?
- Additive identity
- Zero vector = identity vector
- Frame of reference, starting point, nice and neutral
- Additive identity
- Zintegers?
- Why negatives?
- So you can make zero
- Undo each other, undo a \(+5\)
- Inverse
- \(-a\) and \(a\) are additive inverses
- Why negatives?
- Why Zero?
- That's all we need to get to a group: KBe2020math530refGroups