# Mathematical object

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A mathematical object is, loosely speaking, anything you can "do mathematics on". More formally, it is an object that has a definition, obeys certain properties, and can be the target of certain operations.

For example, consider the mathematical object called a complex number:

• Definition:
A complex number is any number that can be written in the form $a+bi$, where $a$ and $b$ are real numbers and the entity $bi$ is an imaginary number, for which the imaginary unit $i^2=-1$.
• Example property:
If $b=0$, then $a+bi$ is a real number.
• Example operation:
Multiplication: $(a_1+b_1i)(a_2+b_2i)=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i$

Note that one mathematical object can be defined in terms of other mathematical objects. In fact, apart from a small number of undefined objects, the existence of which is simply assumed, all mathematical objects are defined in terms of others.

The various areas of mathematics (or mathematical school subjects) can be organized by the type of mathematical object they primarily concentrate on.

Object Subjects or areas of mathematics
Number Arithmetic, number theory
Variable Elementary algebra
Function Intermediate algebra, precalculus, calculus (and higher forms of analysis)
Algebraic structure Abstract algebra
... ...