Assumptions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Assumptions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Statements accepted as true without proof that form the starting conditions for a mathematical argument or model.

What are we assuming to be true? Everything follows from these starting points.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Conclusions are only valid if assumptions hold. Check assumptions first.

Common stuck point: Hidden assumptions are the most dangerous — they silently limit when a result applies. Always ask "What am I taking for granted?" before applying a theorem.

Sense of Study hint: Write a numbered list of everything you are taking for granted. For each item, ask: 'What if this were false?' If the conclusion breaks, that assumption is critical.

Worked Examples

Example 1

easy
A student models a ball thrown upward with h(t) = 20t - 5t^2 (height in metres, time in seconds). List three assumptions embedded in this model.

Solution

  1. 1
    Assumption 1: Air resistance is negligible — the model uses only gravity, not drag forces.
  2. 2
    Assumption 2: Gravity is constant at approximately 10 m/s² (using g \approx 10, so \frac{1}{2}g = 5).
  3. 3
    Assumption 3: The ball moves vertically only — horizontal motion is not modelled.
  4. 4
    These assumptions simplify the physics to obtain a tractable mathematical model.

Answer

\text{Key assumptions: no air resistance, constant gravity, vertical motion only}
Every mathematical model rests on assumptions that limit its validity. Identifying assumptions clarifies when the model is accurate and when it breaks down.

Example 2

medium
In a proof that \sqrt{2} is irrational, the argument begins: 'Assume \sqrt{2} = p/q in lowest terms with p,q integers.' Identify every assumption made and explain why each is necessary.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A shopkeeper assumes 'each customer buys exactly one item.' Under this assumption, if 50 customers visit, how many items are sold? State one way the assumption could fail.

Example 2

medium
A theorem states: 'For all a, b > 0, \frac{a+b}{2} \ge \sqrt{ab} (AM-GM inequality).' Identify the key assumption and give a counterexample showing the result fails without it.
← Back to Assumptions Practice Mode