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Vague answer ... it depends.
More precisely, it depends on what exactly you want to get done, and to what degree you want it to model the real world. In ultra realistic video games, math is key.
My background is mechanical engineering. On the college level, I have taken 3 courses of calculus, 1 of differential equations, 1 of linear algebra (matrix), and 1 of numerical methods (computer solvers for equations, etc).
Here is how things apply. You must must understand high school algebra to get anywhere in programming. If you have a relation, typically to handle it in a program it must be expressed explicitly, which means if it is in terms of x the x must be able to be removed to the left hand side. 1= 2 + x can become x = -1.
If you intend to do graphics. Geometry is needed. Want something to follow your mouse and have the speed related to how far away it is? You are looking for trig relations.
3D modeling. Can you do this without math? These days yes, short answer. Can you use pre-made collision detection schemes, yes. Can you write your own without some knowledge of calculus, maybe, but you will probably learn a bit while you go and it will take you a lot longer than it would take me.
Finally, differential equations. Lets say I wanted a rope to behave like a rope when I picked it up. Or I wanted a car to behave just like a car when I drive it. Or blah blah blah... I want physics. You can hack this together by modifying properties until it "looks right," but if you want it to be truly realistic, some knowledge of differential equations is necessary. This is the bulk of what I do... but I won't bore you with the details as it doesn't directly answer your question.
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