I speak as a PhD in Pure Math, as a director of two companies, one of which employs programmers, and as a coder who writes stuff for myself, and for others.

None of the stuff you mention is essential for a startup. It's likely that almost none of it will be the vaguest bit relevant to your startup before you have to be employing people. If your "startup" fails, or turns into a small side-line, then almost nothing of math or algorithms will be essential.

However, if your startup really does go "viral" and take off in a big way, only then will scaling matter, and at that point the math and algorithms might matter.

Until then, the only math you need is: "How will I make money from this?"

Thanks a lot. However, you do have to know how to program, right? Who else is going to build your product?

You probably will need to be able to program. You probably should know a little about how the web works, and a little about some sort of framework, and something about storing and accessing details in a database.

But all of that is minor to actually getting something running. Anything. Even if you have an idea and validate it by running a service by hand via email. If it starts getting popular (for which read - too much work to do by hand) then you need to automate. At that point you need to be able to code something, anything that works.

Then you need to improve like crazy. Get advice, get help, get a mentor, get a co-founder, or do it yourself.

But you still won't need detailed knowledge of algorithms, calculus, discrete math, etc.

Well, yeah, or your co-founder does.

Or outsource it albeit its much more cost saving if you can do it yourself or find yourself a cofounder. (You used as general case in this sentence).

It depends a lot in the field you'll be working actually. If all your startup does is data ping-pong from fronted to DB, maybe collaboration software or alike, math has little to none value.

If you are building the next speech recognizer, siri 2.0, social media consumer market analysis, meme detection and tracking, financial planning and prediction, quant. trading, ebay bid e-assistant, consumer-advertiser matching etc, then math has a big influence over the result.

It's much worthwhile learning things like the market you're in, psychology, human behavior, and human nature instead of algorithms. The point of a startup is to solve someone's pains and get them to buy it. An algorithm can be part of the execution but it's useless if you don't know what you need to execute, and why. It's like designing a complex financial derivative that ends up destroying Wall Street.

Depends on what your startup will be doing. In general, no. If your startup is doing something like data analysis, then yes you need a lot of math. If you are doing anything involving graphics then a good grasp of trigonometry and linear algebra would be important.

Only one mathematical rule required

Income >= Expenditure

no.