History of mathematics

:See Timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.

Developing the concept of "number" through equations

Many of the extensions of the concept of number can be seen as responses to equations that would otherwise have had no solution. In each of the extensions given below we start with an equation and then give the extension to the system which allows the equation to be solved. We start with the notion of natural numbers: positive integers and zero, although it should be noted that some ancient mathematics did not have the concept of zero. Also note that it was assumed that the normal algebraic operations + - imes / return only one value (division by zero is not defined).

Related Topics:
Natural number - Zero - Division by zero

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• $X+1=0$ requires the existence of negative numbers such as $-1$ for its solution. The word negative was originally used by those who opposed the introduction of such numbers.
• $5\; imes\; X=3$ requires the existence of fractional numbers for its solution. If we allow the solution of all equations of the form $m\; imes\; X=n$ then we get the rational numbers (m and n are both integers).
• $X\; imes\; X-2=0$ has no rational solution. Mathematicians responded by introducing radicals and real numbers, which allowed many polynomial equations to be solved.
• $X\; imes\; X+1=0$ is the equation that introduces us to the complex numbers, which are discussed below.