Add one condition
Special cases and generalisation
A square is a rectangle
A rectangle can have any two sides a and b. A square is a rectangle with an extra condition: its sides are equal, b = a. We say a square is a special case of a rectangle.
The area of a rectangle is ab. What does this give for a square of side a?
What this lesson covers
The idea
A special case results from a general result when an extra condition is applied, and the general result is called a generalisation of the special case.
A square is a rectangle
A rectangle can have any two sides a and b. A square is a rectangle with an extra condition: its sides are equal, b = a. We say a square is a special case of a rectangle.
The area of a rectangle is ab. What does this give for a square of side a?
- a²
- 2a
- 4a
Special cases
Use the three tabs. In each one, apply the extra condition and watch the general result turn into a special case.
General and special
A special case results from a general result when an extra condition is applied. The general result is called a generalisation of the special case.
Put b = a in the area ab of a rectangle and you get a², the area of a square. Put b = a in a² + b² = c² and you get c = a√2 for an isosceles right-angled triangle. Put c = 0 in (a + b + c)² and you get (a + b)². Generalisation is an extremely important part of mathematics.
Notes
A special case comes from a general result when an extra condition is applied. The general result is a generalisation of the special case.
Check yourself
A square of side 5 is a special case of a rectangle. Use the rectangle formula 2(a + b) with b = a = 5. What is the perimeter?
Answer: 20
Perimeter = 2(5 + 5) = 20, which is 4 × 5.
In a right-angled triangle, a² + b² = c². Take the special case a = b = 3. What is c²?
Answer: 18
c² = 3² + 3² = 9 + 9 = 18, which is 2a².
Which statement is correct?
Take the identity for (a + b + c)² and put c = 0. Which identity do you get?
- A = ab is a special case of A = a². It is the other way round: A = ab is the general result.
- A = a² is a special case of A = ab — correct. Yes. Put b = a in A = ab.
- The two formulas are not related. They are related: put b = a in A = ab to get A = a².
- (a + b)² = a² + b². The term 2ab does not have c in it, so it stays.
- (a + b + c)² = a² + b² + c². That is not a result of putting c = 0: with c = 0, the c terms vanish.
- (a + b)² = a² + b² + 2ab — correct. Yes. All the terms with c become 0.