Is it possible to build math backwards?

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In summary: They DO NOT include subtraction or division.In summary, the conversation discusses the concept of doing things backwards in math and how it can be more difficult than doing them forwards. The idea of an "inverse" problem is introduced, where finding the solution is more challenging because there may be multiple solutions or it may not be possible to write a solution in standard ways. The conversation also touches on the idea of reversing the difficulty of processes in math and how it would change the way we learn and think about them.
  • #1
daniel_i_l
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In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.
 
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  • #2
I think you are talking about the "inverse" problem. A "direct" problem, means that we are given a specific forumla, such as "f(x)= x6- 2x5+ 3x+ 4" and asked to evaluate f(2). An "inverse" problem is the other way: "If f(x)= 10, what is x" does not give us a formula. It is more difficult for several reasons: there may be more than one solution (if you did the original problem one of them is easy!), for some polynomials there may be no (real) solution, and, in fact, it may not be possible to write a solution in terms of roots or other standard ways of writing numbers.
 
  • #3
daniel_i_l said:
In many areas of math doing something "backwards" is much harder that doing it "forwards", for example - inverting a matrix is harder than multiplying it, taking a derivative is easier that an integral...
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up. Is it possible to "build math backwards" so that all the things that are hard fo us now are easy and vice versa? If not then why not?
Thanks.
That question reminds one of the arithmetic axioms, which rely ONLY on addition and subtraction. They DO NOT specifically include subtraction or division.
 
  • #4
I think you meant "rely ONLY on addition and multiplication".
 
  • #5
But why should this be, I mean, there's nothing fundamental in the way that we happened to set things up

Yes quite possibly so. Imagine for example that (in some perpendicular universe) things were reversed and integration just happened to be substantially easier than differentiation. Then it's highly likely that integration would then be taught prior to differentiation and differentiation would commonly be thought of as "anti-integration" rather than the other way around. See what's happened, even when we've reversed the relative difficulties your "inverse processes seem harder" observation still holds.
 
  • #6
HallsofIvy said:
I think you meant "rely ONLY on addition and multiplication".
You are exactly correct; I did mean "Addition and Multiplication". The laws of arithmetic which we learn formally during the first two years of "high school algebra" rely only on addition and multiplication.
 

1. What is "Building math backwards"?

"Building math backwards" is a method used in mathematical problem-solving where you start with the solution and work backwards to find the steps or equations needed to reach that solution. It can also be referred to as "working backwards" or "reverse engineering" in math.

2. When is "Building math backwards" useful?

"Building math backwards" can be useful in a variety of mathematical problems, especially in complex ones where it may be easier to start with the solution and work backwards to find the steps or equations needed. It can also be helpful in checking the accuracy of a solution or finding alternative solutions.

3. How do you use "Building math backwards"?

To use "Building math backwards", start with the desired solution and then break it down into smaller, simpler steps or equations. Then, work backwards through each step or equation until you reach the starting point. This can help identify any errors or gaps in the solution and also provide a clearer understanding of the problem.

4. What are the benefits of using "Building math backwards"?

Using "Building math backwards" can help develop critical thinking skills and improve problem-solving abilities. It can also provide a deeper understanding of mathematical concepts and help identify patterns and relationships between different equations or steps.

5. Can "Building math backwards" be used in any type of math problem?

Yes, "Building math backwards" can be used in a variety of mathematical problems, from basic arithmetic to more complex algebraic or geometric problems. It is a versatile method that can be applied to many different types of math problems.

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