12 minute lessonReady to learn

Decide which inputs are valid

Use denominators, logarithms, and comparison spaces to find the domain of a formula.

Course lessons · 2 of 5

Your goal

Identify undefined expressions and explain a valid way to handle their boundaries.

Useful background: Symbols are local definitions

01 · Understand the idea

A formula includes a domain

The domain is the set of inputs for which the stated operations are defined. The archive contains both a bounded reciprocal 1/(1 + δ) and an unregularized reciprocal 1/δ. These are different models: the second is undefined at zero.

For a nonnegative discrepancy, the bounded form lies between zero and one, with one included. A finite discrepancy never produces exactly zero. A programmer should not silently replace an undefined result with zero and then interpret it as a real measurement.

f(\delta)=\frac1\delta,\quad \delta>0

Boundary conventions change the implementation

The expression p log p at p = 0 uses a limiting convention in entropy: its continuous extension is zero. This is a stated mathematical limit, not ordinary evaluation of log zero.

A distance also requires compatible representations. Comparing two positions in the same coordinates is defined; subtracting a sentence from a physical device is not. Before computing a discrepancy, define both representations, their common space, and the distance rule.

02 · Work through an example

Check three discrepancies

1Identify the operation

For 1/δ, inspect the denominator before calculating. A denominator of zero is excluded.

2Evaluate valid cases

For δ = 0.5 and δ = 2, the reciprocal scores are two and one half.

f(0.5)=2,\quad f(2)=0.5

3Handle the excluded case explicitly

For δ = 0, report that the unregularized formula is undefined. Choosing 1/(1 + δ) would define a different bounded score, which must be named.

03 · Change something

Make the idea respond

Use the unity lab to explore the bounded score 1/(1 + δ), where δ = |x − 1|. At x = 1 it returns one because δ is zero. This lab does not implement the singular score 1/δ: that separate expression remains undefined at δ = 0. Compare the two definitions on paper instead of treating one lab output as both formulas.

Change one input at a time. Predict what should happen, then compare the computed result with your prediction.

Teaching model · computed in your browser

Distance from unity

Move the input, watch the distance from 1 change, and calculate the score yourself.

U(x)=\frac{1}{1+|x-1|}

x is a real input and |x − 1| is its absolute distance from 1. The displayed domain is −3 ≤ x ≤ 5; the mathematical function is defined for every real x.

Try this: compare x = 0 and x = 2. Why do they produce the same score?

Experiment controls

Input x 1-3 to 5 · step 0.05

The default worked example is shown below.

Distance |x − 1|0

Unity score1

Maximum1Reached only at x = 1

At x = 1, distance is 0 and the score is 1. The peak is at x = 1.

Substitute, then simplify: U(1) = 1 ÷ (1 + 0) = 1. Equal distances on either side of 1 give equal scores. A score of 1 is a property of this function; it is not evidence of consciousness.

Reference inputs
x	Distance from 1	U(x)
-1	2	0.3333
0	1	0.5
1	0	1
2	1	0.5
3	2	0.3333

Open the full lab

04 · Check your understanding

Your turn.

Try an answer before opening a hint. Feedback will explain the result, and you can practice again.

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View printable practice and solutions

Use these questions when studying without interactive controls. Opening a written solution does not earn progress.

For f(δ) = 1/δ, what is f(0.25)?
Read the solution
The valid positive input gives 1/0.25 = 4.
What should an interface display for 1/δ at δ = 0?
- Undefined under this formula.
- A measured value of zero.
- A measured value of one.
- A guaranteed consciousness event.
Read the solution
The unregularized reciprocal is undefined at zero.
For the bounded teaching score 1/(1 + δ), what value results at δ = 3?
Read the solution
The bounded score is 1/4 = 0.25.
Which comparison has a defined distance after choosing Euclidean distance?
- Two records with all values missing.
- Two position vectors in the same coordinate system and units.
- A paragraph and a physical server with no mapping.
- Meters and seconds subtracted directly.
Read the solution
Two compatible position vectors permit an ordinary Euclidean distance.

05 · Apply it to the archive

Follow the original record

Use the definitions you just practiced to read the project’s own notation, assumptions, and evidence.

Unregularized inverse-discrepancy score Declaration and persistent state change

Take it with you

The ideas to keep

Check domains before evaluation.
A regularizer defines a specific alternative.
Missing and undefined values are not measured zeros.

Reading completion is separate from your practice evidence.

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