Why isn't √1 -i? Understanding the Complex Number System

In this exploration, we delve into the world of complex numbers and the elusive step of why the square root of -1 is the imaginary unit i, but not -i. The fundamental definitions and properties of these numbers provide the clarity.

Definition and Notation

Mathematically, we define the square root of -1 as the imaginary unit i. This is often written as:

i √(-1)

By definition, we can continue to call it √(-1) if we prefer, but we use i as a shorthand for consistency. The letter i stands for "imaginary" (though this has no particular functional significance), being nothing more than an alias for the square root of -1. In physics, you may see the letter j instead.

Historical Context and Misconceptions

The use of i for the square root of -1 has a rich history and has led to some misconceptions. Historically, complex numbers were not well understood, leading to early misunderstandings and naming conventions that still persist today. This is not unique to mathematicians; lay people often misinterpret the names of things due to naming conventions.

For example, the second component of a complex number is named i, which is shorthand for the complex number 01. This is purely a notational convenience and does not imply any deeper meaning. The identification of real numbers x and y with the complex numbers x0 and y0 is just a way to extend the number system.

Complex Number Operations

The operations of addition and multiplication in the complex number system are defined by:

ac bd ac bd (a bi)(c di) ac - bd ad bc

Using these definitions, the square of the complex number 01 (or i) is calculated as follows:

01^2 01 × 01 0 - 100 10 -10

Note that xy x0[01 × y0]. This notation is just a convenient way to represent and operate on complex numbers, with nothing deep or meaningful to it.

Deeper Insight into Square Roots

When you square a number, you are raising it to the power of 2. For real numbers, this operation always results in a non-negative number. However, in the realm of complex numbers, the square root of a negative number, like -1, is i. This is a key point in understanding complex numbers.

For example, if we square the imaginary unit i:

i^2 -1

And since i is defined as √(-1), we can write:

i √(-1)

Some people might be upset to see this, but it is simply a matter of notation. The calculation 01^2 -10 is just as valid, but using i is more familiar and concise.

Pitfalls of Misunderstanding

The question of why √1 is not -i is a common one that stems from a misinterpretation of square roots and complex numbers. It is important to understand that:

(-1^2 -1 × -1 1) If we take the square root of 1, we get 1, not -1.

Therefore, writing √1 -i would be incorrect, as the principal square root of 1 is 1. Squaring both sides of the equation √1 1 gives us 1 1, which is true. However, squaring both sides of √-1 1 leads to a contradiction (since -1 ≠ 1).

To summarize, the square root of -1 is defined as i, but -i is also a valid root. The key is to understand that the principal square root of a number is the non-negative root, unless otherwise specified.