How do calculators have answers for every equation?
In: Mathematics
Calculators store numbers in binary – 1’s and 0’s. To do something like addition, these 1’s and 0’s are compared two at a time. Each comparison takes roughly the time it takes for electricity to flow one millimeter or less. Electricity travels near the speed of light. You can do several of these comparisons at a time. The result is that millions or even billions of tiny comparisons can happen in less than 100 milliseconds (roughly the time it takes for a human to notice delay). Depending on the hardware, as few as two such comparisons are necessary for every binary digit. As such, adding two numbers which are each less than 65,535 can take only 16 comparisons.
Math is like a knot, some knots can be solved quickly while others knots might take longer, but to solve a knot is just to untangle it (fairly easy process). The process of solving the knot is straightforward but harder knots just take longer to solve.
Math is just addition, subtraction, multiplication and division, but sometimes with extra steps the involve the combination of all 4 repeating.
Because that’s all any kind of electronic thing can do–math. Anything else they can do, whether it be a calculator or your smart phone is just a lot of math causing other things to happen. And math is just little virtual switches being turned on and off. This makes addition and subtraction really easy and multiplication and division are just shortcuts for addition and subtraction.
Anything else. is just a more complicated way of doing that, so calculators and computers can do any kind of math you give them.
For basic maths: binary addition, subtraction, and logical shifts
For complex maths (e.g. Trig functions) they use what is called a series expansion that converts the functions into what computer class as basic maths. As stated above
Math boils down a fairly simple handful of rules. You don’t program answers for every problem, you just program the rules. The entirety of math, boils down to less than a megabyte of rules.
Think about how your math teachers explain how to solve a problem. They likely gave you some step-by-step instructions on how to solve for x in equations like y= 2x + 3.
And if we can teach that to students, why not a computer! Many calculators use these step-by-steps (or algorithms), some are very simple to understand and some are quite complex.
math at its most fundamental is an exercise in counting, so it’s not the most difficult thing to do for a computer. it just counts a lot faster then we do.
I’m assuming you’re thinking about a pocket calculator [like this](https://en.wikipedia.org/wiki/TI-30#/media/File:TI-30X_SOLAR,_2.jpg). It doesn’t deal with *equations* at all (what you ask about in your question). It only deals with *expressions* that don’t involve variables.
All the operations (calculator buttons) are things that can be calculated from known variables. If you remember how to add or multiply numbers with pencil and paper, you should realize the step-by-step process [1] can be applied to *any* number regardless of how many digits it has. To add numbers, the hardware and software inside the calculator (which is really a tiny computer) uses the same paper-and-pencil algorithm you would to add them.
More “advanced” buttons like square root, `cos()` or `ln()` also have algorithms which allow them to be calculated. The algorithms are just more complicated.
[1] Mathematicians and computer scientists call a step-by-step process an *algorithm*.
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