tom | Calculus Coaches

October 13, 2025October 13, 2025

Locus of |z − (1 + i)| = 1

What “locus” means: a locus is the set of all points that satisfy a given condition.

We start with the given condition:

|z − (1 + i)| = 1

The “| · |” symbol means the modulus, which is the length of a complex number. This represents its distance from the origin (0) in the Argand plane.

So, |z − (1 + i)| represents the distance from the point z to the fixed point (1 + i) in the complex plane.

The condition |z − (1 + i)| = 1 means all points z that are exactly 1 unit away from the point (1 + i).

To work with this algebraically, represent z using coordinates. Let z be:

z = x + i y

(where x is the real part and y is the imaginary part).

Now substitute this into the original condition:

|(x + i y) − (1 + i)| = 1

Subtract the real and imaginary parts separately:

(x + i y) − (1 + i) = (x − 1) + i (y − 1)

In this expression, the real part is (x − 1) and the imaginary part is (y − 1).

The modulus of a complex number a + i b is given by:

|a + i b| = √(a² + b²)

Apply this formula here with a = (x − 1) and b = (y − 1):

|(x − 1) + i (y − 1)| = √((x − 1)² + (y − 1)²)

The condition states that this modulus equals 1:

√((x − 1)² + (y − 1)²) = 1

Since both sides are non-negative, square both sides to eliminate the square root:

(x − 1)² + (y − 1)² = 1

This is the equation of a circle in the xy-plane with center at (1, 1) and radius 1.

Locus: The set of all points (x, y) satisfying (x − 1)² + (y − 1)² = 1 corresponds to all complex numbers z on this circle, where the distance from z to (1 + i) is 1.

The Argand plane is the complex plane, with the horizontal axis as the real part (Re) and the vertical axis as the imaginary part (Im).

Diagram – Building the Locus Step by Step

August 10, 2025

Δx · f′ ≈ ∑ f′(xᵢ)Δx = ∫ₐᵇ f′(x)dx = f(b) − f(a)

Little slope-triangles along f: base Δx, height Δy ≈ f′(xᵢ)Δx. Summing the vertical legs gives Δf.

f(x) a b Δf = f(b)−f(a)

Graph of f with slope triangles

f′(x) a b

Riemann sum of f′(x)

f(x)

a b

N slicesN = 40 mid-point xᵢ show slope triangles animate

Δx

—

∑ f′(xᵢ)Δx

—

f(b) − f(a)

—

error = |∑ − Δf|

—

What this shows

Pick [a,b] and N, so Δx = (b−a)/N.
On each slice, take xᵢ (midpoint is default).
Δyᵢ ≈ f′(xᵢ)Δx is the little triangle’s vertical leg.
Sum verticals: ∑Δyᵢ ≈ f(b)−f(a). As N↑, equality holds.

Intuition. Over a tiny step Δx the curve is nearly straight, so f(x+Δx) ≈ f(x)+f′(x)·Δx. Slope turns “run” into “rise”. Each triangle’s height is that rise; adding them from a to b gives the net change f(b)−f(a). Errors come from curvature and are ≈(Δx)² per slice, so total error → 0 as Δx → 0.

August 5, 2025August 5, 2025

Refuting Bart Ehrman

52 Points: Ehrman vs. the Resurrection

Below is a detailed table contrasting Bart Ehrman’s common arguments against the resurrection with logical, historical, and metaphysical counterpoints. Each entry expands on the reasoning behind the claim and response to help readers engage more deeply with the issue.

Hallucination Hypothesis
Ehrman: The resurrection appearances were likely grief-induced hallucinations among Jesus’ followers.
Counter: Clinical psychology shows no examples of group hallucinations with consistent content. Hallucinations are highly personal and fragmented — not shared, coordinated, and consistent like the resurrection reports.
Paul’s Vision as Internal
Ehrman: Paul experienced an internal, spiritual revelation, not a physical appearance of Jesus.
Counter: Paul was not grieving — he was a persecutor. His radical change and use of bodily resurrection language (“seen”, not “felt” or “dreamed”) doesn’t match a vague mystical experience.
Disciples May Not Have Died for It
Ehrman: There’s insufficient evidence that disciples died for belief in resurrection.
Counter: Early sources indicate violent persecution of Christians, many of whom explicitly died professing resurrection faith. Even if exact records are sparse, the explosion of faith under persecution is best explained by sincere, core conviction.
Belief ≠ Truth
Ehrman: The disciples may have sincerely believed something false.
Counter: True — sincerity doesn’t prove truth. But sincere belief doesn’t explain the sudden public confidence, the radical moral shift, or the coordinated witness to a physical presence.
Visions Are Common
Ehrman: Many people claim visions; the resurrection is one of many.
Counter: But very few visions — if any — lead to worldwide transformation. The fruit of the resurrection was social reform, ethical clarity, and deep personal change. That’s rare.
Rare Events Are Suspicious
Ehrman: The resurrection is a one-time anomaly — that’s suspicious.
Counter: But so are all foundational events: the Big Bang, emergence of life, consciousness, etc. Rarity doesn’t invalidate reality — it can mark something utterly foundational.
Started in a Remote Place
Ehrman: Christianity just happened to grow from an obscure corner of the empire.
Counter: Movements from obscure villages don’t go global unless something disruptive launches them. A resurrected Jesus explains the anomaly.
Suspicious Timing
Ehrman: The resurrection claim arose just when followers needed hope most.
Counter: The claim wasn’t expected. A crucified messiah shattered expectations — resurrection wasn’t an emotional salve, but a shocking reversal.
Mistaken for Vision
Ehrman: They confused internal experience for external presence.
Counter: The reports describe walking, touching, eating — physical events. These aren’t the hallmarks of a vision or internal impression.
Grief-Induced Delusion
Ehrman: Trauma and grief can cause delusion.
Counter: Trauma usually leads to depression, not radical joy, courage, and world-altering faith. The psychology doesn’t add up.
Greeks Rejected Resurrection
Ehrman: The message was culturally convenient.
Counter: Quite the opposite. Greeks found bodily resurrection disgusting. The gospel offended both Roman pride and Jewish law — that makes it unlikely to be fabricated.
Part of Common Myth Structure
Ehrman: Ancient people told resurrection myths all the time.
Counter: Pagan myths were symbolic and often seasonal. The gospel claims are concrete, public, and anchored in history, not mythology.
Too Convenient
Ehrman: Jesus fits the mold too well.
Counter: He didn’t fit the mold — he redefined it. No one expected a crucified, suffering messiah.
Myth Grows Over Time
Ehrman: The resurrection story evolved slowly.
Counter: 1 Corinthians 15 shows that the core resurrection creed existed within **3–5 years** of the crucifixion. That’s not slow — that’s lightning.
The Gospels Conflict
Ehrman: Differences in the gospel accounts disprove the story.
Counter: Eyewitnesses often disagree on details while agreeing on core facts. In fact, some variation supports authenticity.
Early Christians Lied or Embellished
Ehrman: They had incentive to mythologize.
Counter: But they gained no power — only suffering. People don’t die for a lie they made up.
No External Evidence
Ehrman: There are no Roman records of the resurrection.
Counter: Rome didn’t archive minor sect disturbances. But multiple independent early sources support the core claim.
Only Happens Once
Ehrman: A one-time event can’t be trusted.
Counter: But so is the universe. So is your life. The most important events in reality are singular.
Jesus Fit Expectations
Ehrman: They molded Jesus into a known religious shape.
Counter: Actually, they had to **stretch their worldview** to even accommodate him. He didn’t fit any template.
Delusions Are Chaotic
Ehrman: The resurrection experiences were psychological disorders.
Counter: Delusions are usually chaotic, frightening, and self-absorbed. These experiences produced peace, love, courage, and coherence.
Faith as Wishful Thinking
Ehrman: Belief in resurrection is comforting fantasy.
Counter: The resurrection demanded suffering, self-denial, and martyrdom — not comfort.
We’re More Rational Today
Ehrman: Ancient people were gullible.
Counter: Many were deeply skeptical — Thomas included. They believed because of something overwhelming — not because they were simple-minded.
Only the Desperate Believe
Ehrman: The early believers needed hope.
Counter: They were crushed — and yet suddenly full of joy, mission, and courage. That’s not desperation — it’s transformation.
Modern Science Disallows It
Ehrman: Resurrection violates everything science tells us.
Counter: So does consciousness, free will, and beauty. Reality contains more than formulas. Not everything true is measurable.
Birth Is the Greater Miracle
Ehrman: Resurrection is impossible.
Counter: Your own birth — from nonexistence to consciousness — is more mysterious than rising again. But we’re numb to it because it’s common.
Rationality Rules Out Revelation
Ehrman: If it can’t be repeated, it’s invalid.
Counter: Not true. Unique events are part of how truth unfolds — especially if they reveal more, not less, of what’s real.