prompter: Why can't we divide by 0? Is it just a arbitrary construct created by mathematicians? Couldn't we solve some problems by providing a definition to the division of zero?
assistant: The reason that the result of a division by zero is undefined is the fact that any attempt at a definition lead to a contradiction.

To begin with, how do we define division? The ratio r of two numbers a and b, "r=a/b" is that number "r" that satisfies "a=r*b."

if we are trying to divide by zero, the condition above would be that "b=0", thus we have to find a number r such that "r*0=a", But "r*0=0" for all numbers "r", and so, unless "a=0", there is no possible solution.

Now you could say that "r=infinity" satisfies would satisfy the condition. That's a common way of putting things, however "Infinity" is not a number but a concept, because if we treated "infinity" like a number, more contradictions would arise. 
For example, what do we obtain when adding a number to infinity?

The common perception is that infinity plus any number is still infinity. If that's so, then "infinity = infinity+1 = infinity + 2" which would imply that "1 equals 2" if "infinity" was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

I hope this answers your question.
prompter: Math is confusing. Like those weird Irrational Numbers, how do they even work?
assistant: Irrational numbers can be confusing, but they are actually quite fascinating. An irrational number is a number that cannot be expressed as a ratio of two integers. In other words, it cannot be written as a fraction where both the numerator and denominator are integers.

The most famous irrational number is probably pi (π), which is the ratio of the circumference of a circle to its diameter. Pi is an irrational number because it goes on forever without repeating and cannot be expressed as a fraction.

Another example of an irrational number is the square root of 2 (√2). This number is also irrational because it cannot be expressed as a fraction. If you try to express it as a fraction, you will find that the decimal representation goes on forever without repeating.

Irrational numbers are important in mathematics because they help us to understand the limits of our number system. They show us that there are numbers that cannot be expressed as fractions, and that our number system is not complete. They also have practical applications in fields such as geometry, physics, and engineering.

While irrational numbers can be confusing, they are an important part of mathematics and help us to understand the world around us in a deeper way.

Math is awesome because it is a universal language that allows us to understand and describe the world around us. It is the foundation of science, technology, engineering, and many other fields. Math helps us to solve problems, make predictions, and understand complex systems. It is a tool that allows us to explore the mysteries of the universe, from the smallest particles to the largest structures. Math is also beautiful, with elegant and intricate patterns that can be found in nature, art, and music. It challenges us to think logically and creatively, and it provides a sense of satisfaction when we solve a difficult problem. It is a fundamental part of our lives and our understanding of the world.