Q: What is the science of computing with real numbers called?

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The so-called "real numbers" are simply numbers that permit decimals. Think of them as numbers in scientific notation, which allow very big or very small exponents (somewhere in the range of -300 to +300), and about 15 significant digits. Since their precision is not unlimited, they do not exactly correspond to what is called "real numbers" in mathematics.

real numbers

Real NumbersThe real numbers.

Real and Complex. Real numbers are your everyday numbers that most people are familiar with and Complex numbers are Real numbers mixed with imaginary numbers. Of course this is a VERY BROAD oversimplification, and the question is somewhat vague in that there are other "types" of numbers that are used, but I think this is the answer you are looking for.

Real numbers

The set of real numbers.

A monomial that is a real number is called a constant.

Real numbers are numbers that can be found on the number line. This includes both the rational and irrational numbers. The Real Numbers did not have a name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary.

The Real line.

constant

They are called real numbers. Negative square roots must be complex numbers.

They may be called integers, rational numbers, real numbers. In any one of these sets they are additive inverses.