Number Systems and Proof Techniques: Foundations, Properties, and Logical Reasoning

1. Number Systems: A Hierarchical View

• Closure: Adding or multiplying two elements of the system yields another element of the same system.
• Associativity & Commutativity: Both addition and multiplication are associative and commutative in all these systems.
• Distributive Law: $a(b+c)=ab+ac$ holds universally.
• Identity Elements: $0$ for addition, $1$ for multiplication.
• Inverses: Additive inverse exists in all; multiplicative inverse exists in all except $0$.
• Order: Only $\mathbb{Z},\mathbb{Q},\mathbb{R}$ have a total order.
• Density: Between any two distinct reals there is a rational and an irrational.
• Completeness: Every Cauchy sequence of reals converges to a real; this is not true for rationals or integers.

1.1 Visualizing the Number Line

The diagram shows integers as full ticks, rational non‑integer points as short ticks, and two irrational points marked explicitly.

1.2 Density of Rationals and Irrationals

The points $\frac{a+b}{2}$ and $\frac{a+b}{3}$ are rational when $a,b$ are rational, illustrating that between any two reals there is a rational. A similar construction with $\sqrt{2}$ shows the existence of irrationals between any two reals.

2. Proof Techniques

Mathematical proofs are logical arguments that establish the truth of a statement. Four fundamental techniques are:

1. Direct Proof – Assume the premises and deduce the conclusion using known facts.
2. Proof by Contradiction – Assume the negation of the statement, derive a contradiction, and conclude the statement is true.
3. Contrapositive – Prove $\neg Q⟹\neg P$ to establish $P⟹Q$.
4. Mathematical Induction – Prove a base case and show that if the statement holds for $k$, it holds for $k+1$.

2.1 Proof by Contradiction – A Flowchart

flowchart TD
    A["Assume statement S is false"] --> B["Derive logical consequences"]
    B --> C["Reach contradiction (false statement)"]
    C --> D["Therefore S must be true"]

2.2 Induction – A Flowchart

flowchart TD
    A["Base Case: n=1"] --> B["Assume P(k) true"]
    B --> C["Prove P(k+1) using P(k)"]
    C --> D["Conclusion: P(n) holds for all n≥1"]

3. Worked Examples

3.1 Proving $\sqrt{2}$ is Irrational (Contradiction)

Statement: $\sqrt{2}$ cannot be expressed as a fraction $\frac{p}{q}$ in lowest terms.

1. Assume $\sqrt{2}=\frac{p}{q}$ with $\gcd (p,q)=1$.
2. Square both sides: $2=\frac{p^2}{q^2}$ → $p^2=2q^2$.
3. Thus $p^2$ is even ⇒ $p$ is even ⇒ $p=2k$.
4. Substitute: $(2k{)}^2=2q^2$ → $4k^2=2q^2$ → $q^2=2k^2$.
5. Hence $q^2$ is even ⇒ $q$ is even.
6. Both $p$ and $q$ are even, contradicting $\gcd (p,q)=1$.
7. Therefore, $\sqrt{2}$ is irrational.

3.2 Sum of the First $n$ Natural Numbers (Induction)

Claim: $S_n=1+2+\cdots +n=\frac{n(n+1)}{2}$.

1. Base case $n=1$: $S_1=1=\frac{1\cdot 2}{2}$. Holds.
2. Inductive step: Assume $S_k=\frac{k(k+1)}{2}$.
3. Then $S_{k+1}=S_k+(k+1)=\frac{k(k+1)}{2}+(k+1)$.
4. Factor $(k+1)$: $S_{k+1}=(k+1)\left(\frac{k}{2}+1\right)=(k+1)\frac{k+2}{2}=\frac{(k+1)(k+2)}{2}$.
5. Thus the formula holds for $k+1$.
6. By induction, the formula is true for all $n\ge 1$.

3.3 Sum of Squares Formula (Direct Proof)

Claim: $1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$.

A concise proof uses the identity
$$\sum _{k=1}^n{k}^2=\frac{1}{6}n(n+1)(2n+1),$$ which can be derived by telescoping the difference of cubes: $$k^3-(k-1{)}^3=3k^2-3k+1.$$ Summing from $k=1$ to $n$ and simplifying yields the desired formula.

4. Complex Numbers – Geometry and Applications

Complex numbers $z=a+bi$ can be represented on the complex plane.

Applications

• Signal Processing: Complex exponentials $e^{i\theta }$ model sinusoidal signals in mobile networks.
• Control Systems: Poles and zeros of transfer functions are complex numbers.

5. In the Real World

Worked Real‑World Example – Bank Interest
A bank offers a simple interest rate of $5\%$ per annum on a deposit of $P=10,000$ NPR.
The interest after $t$ years is $I=P\times r\times t$.
For $t=3$ years:
$$I=10,000\times 0.05\times 3=1,500\text{ NPR}.$$ The calculation uses rational numbers ($0.05=\frac{5}{100}$) and real numbers for the final amount.

6. Exam Tip

• Understand the proof structure: State the hypothesis, use definitions, and conclude clearly.
• Label each step: In induction, write “Base case”, “Inductive hypothesis”, and “Inductive step”.
• Check for hidden assumptions: When proving irrationality, ensure the fraction is in lowest terms.
• Practice with past questions: The unit frequently asks for proofs of irrationality, induction on sums, and properties of number systems.
• Time management: Allocate 5–7 minutes for each proof; write concise, logical steps.