For any complex number z the minimum value of |z|+|z-1| is 1.

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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 expression $|z| + |z-1|$ represents the sum of distances from a complex number $z$ to the points $0$ and $1$ in the complex plane. By the triangle inequality, this sum is always greater than or equal to the distance between $0$ and $1$ , which is $1$ . This minimum value of $1$ is achieved when $z$ lies on the line segment connecting $0$ and $1$ . Therefore, the minimum value is $\boxed{1}$ .

The problem asks us to find the minimum value of the expression $|z| + |z-1|$ for any complex number $z$ . This is a classic application of the geometric interpretation of complex numbers and the triangle inequality.

Concept and Intuition: Complex Number Geometry

A complex number $z = x + iy$ can be visualized as a point $(x, y)$ in the complex plane.

The modulus $|z|$ represents the distance of the point $z$ from the origin $(0,0)$ .

Similarly, $|z-w|$ represents the distance between the complex numbers $z$ and $w$ .

In our problem:

$|z|$ is the distance from $z$ to the origin (the point $0$ ).
$|z-1|$ is the distance from $z$ to the point $1$ (which is $(1,0)$ on the real axis).

So, the expression $|z| + |z-1|$ represents the sum of the distances from a point $z$ to two fixed points: $0$ and $1$ .

Consider three points in the complex plane: $A$ , $B$ , and $C$ . The distance between $A$ and $C$ is $|A-C|$ . The distance between $A$ and $B$ is $|A-B|$ , and between $B$ and $C$ is $|B-C|$ . The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. In terms of complex numbers, this means:

z_1

Equality holds if and only if $z_1$ and $z_2$ have the same argument, meaning they lie on the same ray from the origin (i.e., $z_1 = k z_2$ for some non-negative real number $k$ ). Geometrically, this means the three points are collinear, with $B$ lying on the line segment $AC$ .

Let's apply this understanding to our problem.

Step-by-step Solution:

Identify the fixed points and the expression:

We are looking for the minimum value of $|z| + |z-1|$ .

Let $P$ be the point representing $z$ .

Let $A$ be the point representing $0$ .

Let $B$ be the point representing $1$ .

Then $|z|$ is the distance $PA$ , and $|z-1|$ is the distance $PB$ . We want to minimize $PA + PB$ .
Apply the Triangle Inequality to establish a lower bound:

We can rewrite the expression to fit the triangle inequality form.

Consider the complex numbers $z_1 = z$ and $z_2 = -(z-1) = 1-z$ .

Using the triangle inequality $|z_1 + z_2| \le |z_1| + |z_2|$ :

|z + (1-z)| \le |z| + |1-z|

|1| \le |z| + |1-z|

Since $|1| = 1$ and $|1-z| = |-(z-1)| = |z-1|$, we get:

1 \le |z| + |z-1|

This inequality tells us that the sum of the distances $|z| + |z-1|$ is always greater than or equal to $1$. Therefore, the minimum value cannot be less than $1$. …

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Q.For any complex number zzz the minimum value of ∣z∣+∣z−1∣|z|+|z-1|∣z∣+∣z−1∣ is 1.

Concept and Intuition: Complex Number Geometry

Step-by-step Solution:

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Q.For any complex number $z$ the minimum value of $|z|+|z-1|$ is 1.

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