MathFoundations10: Arithmetic with fractions

We define addition and multiplication of fraction to parallel the operations for natural number quotients. A crucial step is to check that these operations are actually well-defined, that is that they respect the notion of equality built into the definition of a fraction.

Category:

Science & Technology

Tags:

• Foundations
• mathematics
• fractions
• operations
• addition
• multiplication
• equality
• Wildberger

[njwildberger]

Hi meichenl,

Glad you are enjoying the series, and I hope you keep watching. I have no objection at all to sets--as long as we use that word in the proper fashion: that is meaning a finite collection of mathematical objects that we can concretely specify.

It is `infinite sets' that are the big problem. All advantages seemingly obtained by talking about such abstractions are illusory and ultimately interfere with understanding mathematics properly. A point I will discuss a lot later!

[Wwallace67]

Minor correction (I stopped watching at 3:22):

You assert as a basic fact that if a*b=a*c then b=c, but consider a=0, b=2, c=3. Does 2=3? I think you need to clarify your basic fact for a/=0.

[njwildberger]

I defined a fraction to be an ordered pair of natural numbers. A natural number was one of 1,2,3,.... So the possibility that a=0 does not arise.

Note that we introduced the symbol 0 in the Hindu-Arabic notational system as an aid to represent natural numbers. However 0 is not itself a natural number.

[Kevin1is7Lucky]

Thank you very much. You are very clear. I blame myself for not practicing basic math, which ultimately weakened my foundation. I always find myself stumped when textbooks make a leap from one equation to the next using simple arithmetic rules. I hope the educational system is revamped so that they focus on building strong foundations in elementary onward. Thanks again. Great series.

[Adnilrocks101]

lol is that a drumstick?

[VivekFerr]

Thank you thank you thank you so very much. I so needed to brush up on my basic math. Am really grateful. Keep up the good work.

[meichenl]

Thanks, Professor. I'll stay tuned then!

[meichenl]

I'm enjoying the presentation for its excellent clarity of thought. However, I don't understand the animosity towards sets. This development of mathematics doesn't use just numbers. It also uses ordered pairs and relations, for example, and presumably will continue adding new mathematical objects as we continue.

The nice thing about set theory from a foundational point of view is that there are only sets - the numbers, ordered pairs, equivalence classes, etc are all just sets.

[Wwallace67]

Good point.

[rubixcubesolve]

interesting