2 is not a complex number.

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Q. 2 is not a complex number.

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Concept understanding — Complex Number Arithmetic

Complex Number Arithmetic: A First Look

Imagine you're trying to solve $x^2 + 1 = 0$ . You know that no real number squared gives $-1$ . The square of any real number is either zero or positive. So this equation has no real solution. But what if we invent a number whose square is $-1$ ? That's exactly what mathematicians did — and that invention is the imaginary unit $i$ , defined by:

i^2 = -1

A complex number is any number of the form $a + bi$ , where $a$ and $b$ are real numbers. Here $a$ is called the real part, and $b$ is called the imaginary part. For example, $3 + 4i$ has real part $3$ and imaginary part $4$ .

Note

The name "imaginary" is unfortunate — these numbers are just as real (in the mathematical sense) as the numbers you already know. They're simply a different kind of number.

Why Bother?

Complex numbers let you solve equations that real numbers can't. Every polynomial equation — no matter how complicated — has a solution in the complex numbers. This is the Fundamental Theorem of Algebra, and it's one of the most important results in mathematics.

Arithmetic Operations

The rules are straightforward: treat $i$ like a variable, but remember that $i^2 = -1$ .

Addition and Subtraction

Add (or subtract) real parts with real parts, imaginary parts with imaginary parts.

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Example: $(2 + 3i) + (4 - 5i) = (2 + 4) + (3 - 5)i = 6 - 2i$

Multiplication

Multiply like binomials, then replace $i^2$ with $-1$ .

(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i

Example: $(2 + 3i)(4 - 5i) = 2(4) + 2(-5i) + 3i(4) + 3i(-5i)$

$= 8 - 10i + 12i - 15i^2$

$= 8 + 2i - 15(-1)$

$= 8 + 2i + 15 = 23 + 2i$

Watch out

The most common mistake: forgetting that $i^2 = -1$ , not $1$ . Always check your final step.

Division

Division is trickier. The key idea: multiply numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of $a + bi$ is $a - bi$ . When you multiply a complex number by its conjugate, you get a real number:

(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 + b^2

So to divide:

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(a + bi)(c - di)}{c^2 + d^2}

Example: $\frac{2 + 3i}{4 - 5i} = \frac{(2 + 3i)(4 + 5i)}{(4 - 5i)(4 + 5i)} = \frac{8 + 10i + 12i + 15i^2}{16 + 25} = \frac{8 + 22i - 15}{41} = \frac{-7 + 22i}{41} = -\frac{7}{41} + \frac{22}{41}i$

Tip

To divide quickly: multiply top and bottom by the conjugate of the denominator, then simplify. The denominator always becomes $c^2 + d^2$ , a positive real number.

The Big Picture …

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The statement "2 is not a complex number" is false. Every real number, including 2, is a complex number of the form $a + 0i$ , where $a$ is real. So 2 is indeed a complex number.

The Concept: What Makes a Number "Complex"?

The confusion here often comes from the word "complex" — it sounds like it means "complicated" or "non-real". But in mathematics, a complex number is simply any number that can be written in the form:

a + bi

where $a$ and $b$ are real numbers, and $i = \sqrt{-1}$ .

$a$ is called the real part.
$b$ is called the imaginary part (note: $b$ itself is real; the "imaginary" part is $bi$ ).

Now, here's the key insight: there is no rule that says $b$ cannot be zero. If $b = 0$ , then the number is $a + 0i = a$ , which is just a real number. So every real number is automatically a complex number — it's a special case where the imaginary part is zero.

Tip

Think of complex numbers as an extension of real numbers, just like real numbers are an extension of integers. The set of real numbers $\mathbb{R}$ is a subset of the set of complex numbers $\mathbb{C}$ . So if someone says "2 is not a complex number", they are essentially saying "2 is not a real number" — which is obviously false.

Step-by-Step Reasoning

Recall the definition of a complex number.

A complex number is any expression of the form $x + yi$ , where $x$ and $y$ are real numbers, and $i^2 = -1$ . The set of all such numbers is denoted $\mathbb{C}$ .
Identify the number 2 in this form.

We can write 2 as $2 + 0i$ . Here, $x = 2$ (real part) and $y = 0$ (imaginary part). Both 2 and 0 are real numbers, so this fits the definition perfectly.
Check the condition for being a complex number.

The definition does not require $y \neq 0$ . It only requires that $x$ and $y$ be real. Since $0$ is a real number, $2 + 0i$ is a valid complex number. …

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Q.2 is not a complex number.

The Concept: What Makes a Number "Complex"?

Step-by-Step Reasoning

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