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Why is e number is in financial math

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Because the definition of the number e lines up with how interest is calculated.

For example, if you have an interest rate of 1%, the value after say 1 period (let's say 1 year) would be 1.01.

If the interest is compounded twice a year you'd have (1+0.01/2)\^2

12 times a year would be (1+0.01/12)\^12

Assuming you can compound continuously, one may wonder if you'd end up with "infinite money". Turns out it is not the case, you'd have to compute the limit as n tends to ininity of (1+0.01/n)\^n which is precisely e\^0.01. You can derive that result yourself by using l'Hospital rule.
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Any (real) function whose rate of increase at any moment is the same as the value it takes at that moment must be a multiple of e^(t), making e pretty much ubiquitous when there are continually changing systems whose state affects its rate of change, like a feedback loop.
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Besides the other answers, one of the reasons e (or really, the exponential function) shows up in the Black-Scholes formula, is because it is part of the probability density function of a normal random variable.

Edit: and of course discounting
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There’s an inherent relationship between e and compound interest other have explained.

There’s also a consistent ease of use using exponential/log rules. (1.01)^100 = e^ln(1.01)100 which is in line with the standard formula e^kt for compound interest even though you’re not compounding continuously.

Finally, for frequent long term compounding, like your example, the answers are approximately the same. Taking your interest rate as .01, you answer 1.01^100 = 2.7048 ~ e^(.01x100)= e

I’m sure there are better reasons from specialists , but these are just some observations from teaching the intro to this topic a few times.
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It's a lot easier to perform calculus on exponential functions that are expressed in terms of e than in any other base. And calculus is useful for things like calculating the rate of change of your graph of over time.

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