Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Distinguish Euclidean and taxicab metrics through their geometric balls.
- Justify the conclusion "||(x,y)||₂≤||(x,y)||₁≤√2||(x,y)||₂" using the stated assumptions.
Before you start
Absolute values and coordinates.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Describe the points at distance at most one from the origin under d₁ and d₂.
Why this math matters
Distinguish Euclidean and taxicab metrics through their geometric balls. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The space is R².
- Both metrics use the same coordinate units.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Compare distance rules and their unit balls
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Describe the points at distance at most one from the origin under d₁ and d₂.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
d₁((x,y),0)=|x|+|y|; d₂((x,y),0)=√(x²+y²)
Different metrics assign different costs to diagonal motion.
Work through the mathematics
d₁≤1 gives a diamond; d₂≤1 gives a disk
The boundary equations identify the two unit-ball shapes.
Check the conclusion
||(x,y)||₂≤||(x,y)||₁≤√2||(x,y)||₂
Squaring and using 2|xy|≤x²+y² proves that the two distances control each other.
The result
||(x,y)||₂≤||(x,y)||₁≤√2||(x,y)||₂
Squaring and using 2|xy|≤x²+y² proves that the two distances control each other.
Common mistakes to catch
- A ball need not look circular in the usual drawing.
- Equivalent convergence does not mean equal numerical distances.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Compute both distances from (0,0) to (3,4).
Show a hint
Sum absolute values, then use Pythagoras.
Reveal answer and explanation
d₁=7 and d₂=5
Taxicab motion follows coordinate directions while Euclidean distance uses the direct segment.
Practice 2
Does changing to d₁ change which sequences converge in R²?
Show a hint
Use the two-sided norm bounds.
Reveal answer and explanation
No
A distance tends to zero in one metric exactly when it does in the other.
Take the idea with you
Compare walking through a rectangular street grid with straight-line travel.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Separate translation from a linear transformation
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.