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Number and Algebra Practice (Interactive)

Intuition

Number theory is the detective work of mathematics — it investigates the hidden properties of integers and primes: Algebraic structures like groups, rings, and fields reveal deep symmetries in mathematical systems

Why it matters: From cryptography to coding theory to abstract algebra, number theory connects pure mathematics to real-world applications

The key insight: Algebraic structures like groups, rings, and fields reveal deep symmetries in mathematical systems

IB Maths — Number and Algebra Practice

16 auto-graded practice problems covering core IB Maths number and algebra content, aligned to the SL/HL syllabus. Select an answer, submit, and review the explanation.

Worked Examples

Example 1: Complex Number Division

Problem: Simplify 3+2i1−i\frac{3 + 2i}{1 - i}.

Solution: Step 1: Multiply numerator and denominator by the conjugate of the denominator: 3+2i1−i×1+i1+i\frac{3 + 2i}{1 - i} \times \frac{1 + i}{1 + i}

Step 2: Numerator: (3+2i)(1+i)=3+3i+2i+2i2=3+5i−2=1+5i(3 + 2i)(1 + i) = 3 + 3i + 2i + 2i^2 = 3 + 5i - 2 = 1 + 5i

Step 3: Denominator: (1−i)(1+i)=1−i2=1−(−1)=2(1 - i)(1 + i) = 1 - i^2 = 1 - (-1) = 2

Step 4: Answer: 1+5i2=12+52i\frac{1 + 5i}{2} = \frac{1}{2} + \frac{5}{2}i

Key insight: Always multiply by the conjugate to make the denominator real. The conjugate of a+bia + bi is a−bia - bi.

Example 2: Geometric Sequence

Problem: Find the sum of the first 6 terms of the geometric sequence 5, 15, 45, …

Solution: Step 1: First term a=5a = 5, common ratio r=155=3r = \frac{15}{5} = 3

Step 2: Sum formula: Sn=arn−1r−1S_n = a \frac{r^n - 1}{r - 1}

Step 3: S6=5×36−13−1=5×729−12=5×7282=5×364=1820S_6 = 5 \times \frac{3^6 - 1}{3 - 1} = 5 \times \frac{729 - 1}{2} = 5 \times \frac{728}{2} = 5 \times 364 = 1820

Key insight: The geometric sum formula works for any r≠1r \neq 1. For ∣r∣<1|r| < 1, the sum to infinity is S∞=a1−rS_\infty = \frac{a}{1 - r}.

Example 3: Proof by Induction

Problem: Prove that ∑r=1nr=n(n+1)2\sum_{r=1}^{n} r = \frac{n(n+1)}{2} for all n∈Z+n \in \mathbb{Z}^+.

Solution: Step 1 (Base case): n=1n = 1: LHS =1= 1, RHS =1(2)2=1= \frac{1(2)}{2} = 1 ✓

Step 2 (Inductive hypothesis): Assume true for n=kn = k: ∑r=1kr=k(k+1)2\sum_{r=1}^{k} r = \frac{k(k+1)}{2}

Step 3 (Inductive step): Show true for n=k+1n = k+1: ∑r=1k+1r=∑r=1kr+(k+1)=k(k+1)2+(k+1)\sum_{r=1}^{k+1} r = \sum_{r=1}^{k} r + (k+1) = \frac{k(k+1)}{2} + (k+1) =k(k+1)+2(k+1)2=(k+1)(k+2)2= \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}

This equals the formula with n=k+1n = k+1. ✓

Key insight: In the inductive step, factor out (k+1)(k+1) first — it makes the algebra cleaner.

Number Sets and Operations

Sequences and Series

Exponentials and Logarithms

Proof

Cross-References

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.