KinderCalculus/forever numbers
The terms "infinity" & "infinitesimal" are quite a mouthful. The term "forever" is already a part of young children's vocabulary, so I use "forever big" & "forever small" instead.
We roughly follow Kiesler's text Non-Standard Calculus, but do not go into the details of the Extension and Transfer Principles. They are intuitive and children have been exposed to them since Kindergarten, when the number system undergoes successive extensions: Naturals -> Integers -> Rationals -> Reals -> Complex. After these extensions, yet another extension to Forever Big and Forever Small is not a big leap.
Below we summarize Chapter 1 of Keisler's text. As a matter of notation, ε & δ will be infinitesimals, H & K will be the infinite numbers, and a & b will be reals.
Definitions[edit | edit source]
- Hyperreals is an Extention of the real numbers that includes ±infinity, ±infinitesimal, and numbers of the form finite + infinitesimals. Statements about the reals Transfer to the hyperreals.
- Infinitesimals are zero, ±ε
- finite numbers are those that can be bounded by a real number, including those of the form c+ε
- Infinite numbers are of the form ±H
Infinity[edit | edit source]
Understanding Infinity is one of mankind's greatest achievements. It's a simple idea that every child has asked: "is infinity + 1 bigger than infinity?" and the answer is surprisingly understandable by the same child. No big fancy arithmetic or algebra is needed!
We go over Geogr Cantor's definition of size comparison. "Matching" is our term for the 1-to-1 & Onto mapping of sets. We go over the diagonal argument for irrationals, and power sets for more infinities. Here's is a good video http://ed.ted.com/lessons/how-big-is-infinity
Hyperreal Arithmetic[edit | edit source]
from page 31 of Keisler's text:
infinitesimal | infinite | finite | indeterminate |
---|---|---|---|
1 + H, ε + δ | 1/ε, H + ε, H + b | b + ε | H + K |
ε * δ | H * K, H * b | . | H * ε |
ε/b, e/H, b/H | b/ε, H/ε, H/b | . | H/K, ε/δ (eg. ε^{2}/ε, ε/ε, ε/ε^{2}) |
^{n}√ε | . | . |
Standard-Parts Arithmetic[edit | edit source]
The Standard part of a hyperreal number, st(b), is the closest real number to b. We call it the best real estimate. Hence, st(c) - c = ε. The standard part is well defined & unique. Starndard Part arithmetic, from page 37:
- st(-a) = -st(a)
- st( a+b ) = st(a) + st(b)
- st( a-b ) = st(a) - st(b)
- st( ab ) = st(a) * st(b)
- st( a/b ) = st(a) / st(b), for non-zero st(b)
- st(a^{n}) = (st(a))^{n}
- st( ) = , for positive a
- st(a) <= st(b)
Exercises[edit | edit source]
Section 1.5 (page 34) & Section 1.6 (page 40)