SL 1.1—Using standard form

SL 1.2—Arithmetic sequences and series

SL 1.3—Geometric sequences and series

SL 1.4—Financial apps – compound interest, annual depreciation

SL 1.5—Intro to logs

SL 1.6—Simple proof

SL 1.7—Laws of exponents and logs

SL 1.8—Sum of infinite geo sequence

SL 1.9—Binomial theorem where n is an integer

AHL 1.10—Perms and combs, binomial with negative and fractional indices

AHL 1.11—Partial fractions

AHL 1.12—Complex numbers – Cartesian form and Argand diag

AHL 1.13—Polar and Euler form

AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers

AHL 1.15—Proof by induction, contradiction, counterexamples

AHL 1.16—Solution of systems of linear equations

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

Simple Deductive Proof

Simple deductive proof is a fundamental concept in mathematics that involves demonstrating the truth of a statement through logical reasoning.

At SL level, proofs will be limited to showing that one side of an equation is equal to another.

Theory of Knowledge

In mathematics, new knowledge can only be derived from preexisting knowledge through proofs. What does this mean for the validity of mathematical knowledge?

Numerical Proofs

Numerical proofs involve demonstrating the truth of a mathematical statement using specific numbers.

These proofs are often used to verify simple arithmetic relationships.

Example question

Prove that $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$.

Solution

Start with the left-hand side: $\frac{1}{4} + \frac{1}{12}$
Find a common denominator: $\frac{3}{12} + \frac{1}{12}$
Add the fractions: $\frac{4}{12}$
Simplify: $\frac{1}{3}$
Thus, we have shown that $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$

Algebraic Proofs

Algebraic proofs involve demonstrating the truth of a mathematical statement using variables and algebraic manipulations.

These proofs are more general and apply to a wider range of cases.

Example question

Prove that $\frac{1}{m+1} + \frac{1}{m^2+m} ≡ \frac{1}{m}$ for all $m ≠ 0, -1$.

Solution

Start with the left-hand side: $\frac{1}{m+1} + \frac{1}{m^2+m}$
Find a common denominator: $\frac{m^2+m}{(m+1)(m^2+m)} + \frac{m+1}{(m+1)(m^2+m)}$
Add the fractions: $\frac{m^2+2m+1}{(m+1)(m^2+m)}$
Factor the numerator: $\frac{(m+1)^2}{(m+1)(m^2+m)}$
Cancel $(m+1)$ from numerator and denominator: $\frac{m+1}{(m^2+m)}$
Factor out $m$ from the denominator: $\frac{m+1}{m(m+1)}$
Cancel $(m+1)$ from numerator and denominator: $\frac{1}{m}$
Thus, we have shown that $\frac{1}{m+1} + \frac{1}{m^2+m} ≡ \frac{1}{m}$ for all $m ≠ 0, -1$

Note

The symbol ≡ is used to denote an identity, which means the equation is true for all values of the variable (except where undefined).
This is different from the equality symbol =, which may only be true for specific values.

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IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

Simple Deductive Proof

Numerical Proofs

Algebraic Proofs

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Simple Deductive Proof

Numerical Proofs

Algebraic Proofs

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Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

Simple Deductive Proof

Numerical Proofs

Algebraic Proofs

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Simple Deductive Proof

Numerical Proofs

Algebraic Proofs

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics