Why do we need the properties of numbers? How do they help us in mathematics? Do properties help us remember certain tricks? Is that the only reason for the properties?Why do the properties of numbers exist?
Which properties are you talking about in particular? The basic axioms of algebra?Why do the properties of numbers exist?
We need the properties of numbers in order to figure out other things about them. For instance, we need to know that basic operations like addition and subtraction obey the closure property over the real numbers.
If we didn't know that, it would be pretty difficult to solve high-school algebra problems.
Similarly, if we didn't know that the principle of mathematical induction applied to the natural numbers, many proofs would break down.
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