# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-numbers-proofhl.html


## Proofs

"Pure mathematics is, in its way, the poetry of logical ideas." Albert Einstein


#### Proof by induction


##### Prior Knowledge

- Recognize all the basic functions
- You must be able to manipulate algebraic expressions. (If not, check this lesson)
- You must be able to manipulate exponential expressions. (If not, check this lesson)
- You must be able to manipulate logarithmic expressions. (If not, check this lesson)

##### Key points

- 1. Watch the video:
- 2. Read this presentation
- 3. Watch this solved problem

#### Proof by contradiction


##### Prior Knowledge

- To manipulate algebraic expressions with exponents

##### Key Points

- 1. Watch the video:
- 2. Read this presentation

#### Extra


##### Material and references:

- Hodder Book HL(ISBN: 9781510462366) : 5A, 5B, 5C

##### Key Points

- 1. You should be able to use the principle of mathematical induction to prove statements about patterns involving integers. If you can show that the statement is true for \( n = 1 \), and that when the statement is true for \( n = k \) it is also true for \( n = k + 1 \), then the statement is true for all positive integers \( n \). The proof can be adapted for a starting value other than \( n = 1 \).
- 2. You should be able to use proof by contradiction: assume that the statement you are trying to prove is false and show that this leads to an impossible or contradictory conclusion.
- 3. You should be able to use counterexamples to disprove a statement. One counterexample is sufficient to prove that a statement is false.

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