Trap a root before approximating it
What can a shrinking interval prove about an approximate answer?
Entering midway? A root is an input where a function equals zero. A continuous function with opposite signs at two endpoints crosses zero between them.
Start with a concrete question
Approximate the nonnegative square root of a target by bisecting a valid bracket for x²−target.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
After four midpoint cuts, how much smaller should the bracket be?
Change one thing. Inspect what happens.
- Trace four cuts and verify the selected half still brackets the root.
- Increase the iteration limit and compare residual with the location-error bound.
- Choose endpoints that miss the root and explain why the model refuses to give a certified approximation.
- Try a root at an endpoint and target zero; check exact termination.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Root and requested bracket
Horizontal: x. Vertical: x² − target.
- 1. Function (solid line)
- 2. Zero (dashed line)
- 3. Initial bracket endpoints (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | x² − target |
|---|---|---|
| Function | 0 | -2 |
| Function | 0.03333 | -1.999 |
| Function | 0.06667 | -1.996 |
| Function | 0.1 | -1.99 |
| Function | 0.1333 | -1.982 |
| Function | 0.1667 | -1.972 |
| Function | 0.2 | -1.96 |
| Function | 0.2333 | -1.946 |
| Function | 0.2667 | -1.929 |
| Function | 0.3 | -1.91 |
| Function | 0.3333 | -1.889 |
| Function | 0.3667 | -1.866 |
| Function | 0.4 | -1.84 |
| Function | 0.4333 | -1.812 |
| Function | 0.4667 | -1.782 |
| Function | 0.5 | -1.75 |
| Function | 0.5333 | -1.716 |
| Function | 0.5667 | -1.679 |
| Function | 0.6 | -1.64 |
| Function | 0.6333 | -1.599 |
| Function | 0.6667 | -1.556 |
| Function | 0.7 | -1.51 |
| Function | 0.7333 | -1.462 |
| Function | 0.7667 | -1.412 |
| Function | 0.8 | -1.36 |
| Function | 0.8333 | -1.306 |
| Function | 0.8667 | -1.249 |
| Function | 0.9 | -1.19 |
| Function | 0.9333 | -1.129 |
| Function | 0.9667 | -1.066 |
| Function | 1 | -1 |
| Function | 1.033 | -0.9322 |
| Function | 1.067 | -0.8622 |
| Function | 1.1 | -0.79 |
| Function | 1.133 | -0.7156 |
| Function | 1.167 | -0.6389 |
| Function | 1.2 | -0.56 |
| Function | 1.233 | -0.4789 |
| Function | 1.267 | -0.3956 |
| Function | 1.3 | -0.31 |
| Function | 1.333 | -0.2222 |
| Function | 1.367 | -0.1322 |
| Function | 1.4 | -0.04 |
| Function | 1.433 | 0.05444 |
| Function | 1.467 | 0.1511 |
| Function | 1.5 | 0.25 |
| Function | 1.533 | 0.3511 |
| Function | 1.567 | 0.4544 |
| Function | 1.6 | 0.56 |
| Function | 1.633 | 0.6678 |
| Function | 1.667 | 0.7778 |
| Function | 1.7 | 0.89 |
| Function | 1.733 | 1.004 |
| Function | 1.767 | 1.121 |
| Function | 1.8 | 1.24 |
| Function | 1.833 | 1.361 |
| Function | 1.867 | 1.484 |
| Function | 1.9 | 1.61 |
| Function | 1.933 | 1.738 |
| Function | 1.967 | 1.868 |
| Function | 2 | 2 |
| Function | 2.033 | 2.134 |
| Function | 2.067 | 2.271 |
| Function | 2.1 | 2.41 |
| Function | 2.133 | 2.551 |
| Function | 2.167 | 2.694 |
| Function | 2.2 | 2.84 |
| Function | 2.233 | 2.988 |
| Function | 2.267 | 3.138 |
| Function | 2.3 | 3.29 |
| Function | 2.333 | 3.444 |
| Function | 2.367 | 3.601 |
| Function | 2.4 | 3.76 |
| Function | 2.433 | 3.921 |
| Function | 2.467 | 4.084 |
| Function | 2.5 | 4.25 |
| Function | 2.533 | 4.418 |
| Function | 2.567 | 4.588 |
| Function | 2.6 | 4.76 |
| Function | 2.633 | 4.934 |
| Function | 2.667 | 5.111 |
| Function | 2.7 | 5.29 |
| Function | 2.733 | 5.471 |
| Function | 2.767 | 5.654 |
| Function | 2.8 | 5.84 |
| Function | 2.833 | 6.028 |
| Function | 2.867 | 6.218 |
| Function | 2.9 | 6.41 |
| Function | 2.933 | 6.604 |
| Function | 2.967 | 6.801 |
| Function | 3 | 7 |
| Zero | 0 | 0 |
| Zero | 3 | 0 |
| Initial bracket endpoints | 0 | -2 |
| Initial bracket endpoints | 3 | 7 |
Follow the calculation
- Seek a nonnegative root of x²−2=0 on [0,3].
- Step 1: midpoint 1.50000, residual 0.25000; keep left half.
- Step 2: midpoint 0.75000, residual -1.43750; keep right half.
- Step 3: midpoint 1.12500, residual -0.73438; keep right half.
- Step 4: midpoint 1.31250, residual -0.27734; keep right half.
- Final bracket [1.31250,1.50000], estimate 1.40625; error at most half-width 0.09375 for this continuous monotone function.
The bracket gives an error bound on the root location. A small residual alone is a different quantity and need not imply small location error for every function.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
A root is an input where a function equals zero. A continuous function with opposite signs at two endpoints crosses zero between them.
Approximate the nonnegative square root of a target by bisecting a valid bracket for x²−target.
A worked explanation
Bisection preserves a containing interval. Each nonterminal cut halves its width, and the midpoint is at most half the final width from a contained root.
For √2 on [0,3], midpoint 1.5 gives residual 0.25, so keep [0,1.5]. Midpoint 0.75 is below the root, so keep [0.75,1.5].
The continuity and bracket assumptions matter. A failed bracket means this setup has no certificate; it does not mean the equation has no solution.
Symbols and relationship
Midpoint=(left+right)/2. After k cuts, width≤initial width/2^k; midpoint location error≤width/2 for a valid containing bracket.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Continuous monotone x²−target on the nonnegative domain; the update direction uses that monotonicity.
- Binary64 rounding remains, but at most twelve cuts keep the example far from rounding-limited convergence.
Try a fresh case
A valid bracket starts with width 8. After five nonterminal bisection cuts, what is the midpoint error bound?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Perform four cuts by hand and retain both endpoints.
College investigation
Implement a generic sign-based bisection and compare against Newton iteration with failure cases.
Workplace investigation
Report a root approximation together with its stopping criterion and a valid bound.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.