Why mathematicians invented a number that doesn't exist on a line
In short: For centuries, the square root of a negative number was a mathematical dead end. There is no real number that, when squared, results in a negative value. To break this barrier, mathematicians defined a new kind of number that expands our algebraic universe.
The core motivation for complex numbers is solving equations like x^2 = -1. In standard real-number arithmetic, this is impossible because squaring any real number—whether positive or negative—always yields a non-negative result. To overcome this, mathematicians defined the imaginary unit, i, as the positive solution to x^2 = -1. This is conceptually similar to how the square root of 2 was defined as a solution to x^2 = 2, even when it couldn't be expressed as a simple integer.
A complex number is expressed in the standard form a + bi, where 'a' is the real part and 'bi' is the imaginary part. In this framework, 'a' and 'b' can be any real numbers, including integers, fractions, decimals, or zero. If the real part is zero, the number is called purely imaginary. Conversely, if the imaginary part is zero, the number is simply a real number. This reveals a fundamental hierarchy: real numbers are actually a subset of the broader complex number system.
One of the most significant consequences of this system is the Fundamental Theorem of Algebra. According to this theorem, a polynomial of degree n possesses precisely n complex roots, provided multiplicity is taken into account. This provides a level of mathematical consistency and ease that is not available when working strictly with real numbers. To perform calculations, complex numbers can be treated like polynomials; for instance, addition and subtraction involve combining the real and imaginary parts separately. Additionally, every complex number has a conjugate, formed by flipping the sign of its imaginary part.
By defining i, we can reduce any square root of a negative number into a manageable form. For example, the square root of -9 can be rewritten as 3i. This allows mathematicians to 'make the problem go away' by converting impossible real-number operations into solvable complex-number arithmetic.
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Sources
- Summary by Intellect · source: Intro to COMPLEX NUMBERS // Motivation, Algebraic Definition & Fundamental Theorem of Algebra Ep. 1
- Also: Algebra - Complex Numbers
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