Topics In Applied Math

1.17: The definitions of an ordered field $F$, for $x,y,z \in F$ [i] $x+y \lt x+z$ if $y\lt z$ [ii] $xy \gt 0$ if $x\gt 0 \land y \gt 0$ Review the basic rules of working with inequalities. 1.18.a. Prove $x \gt 0 \implies -x \lt 0$ $$ \begin{array}{l c} x \gt 0 & \\ -x + x \gt -x + 0 & \\ 0 \gt -x & \end{array} $$ 1.18.b. Prove $x\gt 0 \land y \lt z \implies xy \lt xz$ $$ \begin{array}{l c} y \lt z & \\ y-y \lt z-y & \\ 0 \lt z-y & \\ 0x \lt (z-y)x & (?\times x) \ since \ x \gt 0 \\ 0 \lt zx - yx & \\ yx \lt zx -yx + yx & \\ yx \lt zx \end{array} $$ 1.18.c. Prove $x\lt 0 \land y \lt z \implies xy \gt xz$ $$ \begin{array}{l c} y \lt z & \\ y-y \lt z-y & \\ 0 \lt z-y & \\ 0x \lt -(x(z-y)) & \ since \ x \lt 0 \\ 0x \lt (-x)(z-y) & 1.16.c \\ 0 \lt (-x)z + (-x)(-y) & \\ 0 \lt -xz + (-x)(-y) & \\ 0 \lt -xz + (-(-xy)) & \\ 0 \lt -xz + xy & \\ xz \

Imagine you are a certain distance from an objective. Lets say this distance is 16 units. You are allowed to make progress toward your objective by cutting the distance in half at any time, as many times as you like. When will you reach your objective? In order to answer this question we must evaluate a limit. But before doing that, we may ask ourselves another question: how many times can we divide the distance? The distance in this case, at each step, is a rational number. A rational number is a quotient of two whole numbers in the form n/m (hence the number system is denoted “Q”). So our problem is within Q, the set of rational numbers. Mathematically we have distance $d=\frac{16}{2^n}$ where n is our step number. At one step $d=8$, at 2 steps $d=4$, and then $2$,$1$,$\frac{1}{2}$,$\frac{1}{4}$, etc. Our progress, one step at the time, is called a sequence (in Q). A sequence means we can denumerate the step number $n$. We can assign an integer number to each step. It turns ou