Run the Math: Calculus I: Real Numbers, Absolute Values, and Intervals

in runthemath •  2 years ago

Run the Math

Real Numbers, Absolute Values, and Intervals
By Bryan Bartlett
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Real Numbers

Real Numbers (R) are any numbers that can be expressed in decimal form. This includes

Natural Numbers (N): Also known as counting numbers, this would be positive integers {1,2,3,4,5...}
Integers (Z): Which include all of N plus all negative Integers plus 0.
Rational Numbers (Q), which are any numbers that can be written as p/q, where p,q are integers
and Irrational Numbers, which are all non-terminating and non-repeating decimals, like the square root of 2, pi, or e.

Note: For the rest of the lectures, we will use N0 to denote the set of positive integers, with 0. When we want to talk about just the positive integers, we will use Z+. This is due to cultural differences on what the definition of N is, and we are all about keeping things simple

Real numbers are ordered, meaning they have a set, never changing order to them. 2 is always bigger than 1, and pi is always between 3 and 4. Given that real numbers are ordered, we can state the following rules when dealing with any 3 real numbers a,b and c.

If a < b and b < c, then a < c
if a < b, then a ± c < b ± c
if a < b and 0 < c, then a * c < b * c
if a < b and c < 0, then b * c < a * c
if 0 < a < b or a < b < 0, then 1 / b < 1 / a

Note: the rules are the same for >,≤ and ≥

Absolute Values

The absolute value of a number |a| is its magnitude: that is how far away from 0 it is.. regardless of the sign. This means |a| = a if a ≥ 0, and |a| = -a if a < 0. In practice, it means removing the sign.

Extra Credit: Think of the absolute value as the length of a line that goes from 0 to a. This becomes important later because the | | will come up again when dealing with vectors of higher dimensions. If you wanted to find |(x,y)|, you would use √(x2 + y2), which is a positive number! We are doing the same thing in here but in one dimension.. √(a2)
Note: We will assume √ references the positive square root. When I want the negative square root, the convention would be -√

The following are rules that apply to the use of absolute numbers, given that a and b are real numbers

|a| > 0 if a ≠ 0, |0| = 0
|-a| = |a|
|a * b| = |a| * |b|
|a / b| = |a| / |b|, b ≠ 0
|a + b| ≤ |a| + |b| (Triangular Inequality)
|a - b| ≤ |a| + |b|
|a| - |b| ≤ |a - b|

Reminder: If you end up multiplying or dividing by a negative number, the inequality flips around!

Intervals and Neighborhoods

When we write intervals, we use a combination of parenthesis ( ) and Bracket [ ]. They allow us to set a specific group of numbers. Let us assume a ≤ b. If we wanted every number between a and b, but not including a and b, we would use parenthesis to show this, in the form (a,b). If we wanted every number between a and b, including a and b, we would use brackets, in the form [a,b]. You can combine them as well. [a,b) is the interval between a and b, including a but not including b. This mixed format is seen a lot when you have an infinite on one side or another. [a,∞) for example, would be all numbers greater or equal to a.

We can also write the interval in a set notation: (a,b) = {x|a < x < b}, which simply states All x, where x is greater than a and less than b.

Neighborhoods work a bit differently. In this, the segment is centered around a number a, instead of having a at one of the endpoints. An example of this would be {x|a - b< x < a + b} = (a - b,a + b). Otherwise, it works the same.

*Extra credit: A deleted neighborhood is one where a itself is removed from the interval.. [a - b,a) U (a, a + b]. U means union of two sets. since neither of the sets contains a, due to the parenthesis, this is an example of a deleted neighborhood.

Assignment:

  1. Show that 0.125 is a real number using definition of Q
  2. Using the definitions of a real number, show why √-1 is not a real number
  3. -4x < 8 Using bracket notation, what is x?
  4. Write the following in bracket notation: {x | 3a2 - 4 ≤ x < a - b}
  5. Write the following in set notation: (-∞,a + 5)
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