Summary of Animal Reproductive Strategies and Development

## Introduction Numbers are the foundation of mathematics and appear in every aspect of science, technology, finance, and daily life. This study material covers the main types of numbers, how they relate, and how to work with them effectively. ### What you will learn - The different kinds of numbers and their defining properties - How to compare and operate with numbers - Real-world applications of number types and operations > **Definition:** A number is an abstract object used to count, measure, and label. Numbers follow rules that let us combine and compare them. ## 1. Types of Numbers We break numbers into categories so each idea is easier to understand. ### 1.1 Natural numbers > **Definition:** Natural numbers are the counting numbers used for discrete objects: $1,2,3,\dots$ Sometimes $0$ is included. - Notation example: $\mathbb{N} = \{0,1,2,3,\dots\}$ or $\mathbb{N}^+ = \{1,2,3,\dots\}$ - Use: counting items, ordering, indexing ### 1.2 Whole numbers and Integers > **Definition:** Whole numbers include $0$ and positive integers; integers include negative numbers: $\dots,-2,-1,0,1,2,\dots$ - Notation: $\mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$ - Operations: closed under addition, subtraction, multiplication, but not always division ### 1.3 Rational numbers > **Definition:** A rational number is any number that can be written as $\frac{p}{q}$ where $p$ and $q$ are integers and $q\neq 0$. - Examples: $\frac{3}{4}$, $-2$, $0.5$ (which is $\frac{1}{2}$) - Decimal representation: either terminating (e.g., $0.75$) or repeating (e.g., $0.333\dots$) ### 1.4 Irrational numbers > **Definition:** An irrational number cannot be written as $\frac{p}{q}$ with integers $p,q$ and has a non-repeating, non-terminating decimal expansion. - Examples: $\sqrt{2}$, $\pi$ - Important fact: sum of a rational and an irrational number is irrational ### 1.5 Real numbers > **Definition:** Real numbers include all rational and irrational numbers. They correspond to points on a continuous number line. - Notation: $\mathbb{R}$ - Use: measurements, lengths, temperatures ### 1.6 Complex numbers (brief) > **Definition:** A complex number has the form $a+bi$ where $a,b\in\mathbb{R}$ and $i^2=-1$. - Notation: $\mathbb{C}$ - Use: electrical engineering, signal processing ## 2. Number Line and Order - The number line places numbers in order: left is smaller, right is larger - Integers are evenly spaced; rationals and irrationals fill the line densely Fun fact: Every interval on the number line, no matter how small, contains infinitely many rational and infinitely many irrational numbers. ## 3. Properties and Operations ### 3.1 Basic operations - Addition, subtraction, multiplication, division (except by zero) - For integers: addition and multiplication are closed operations ### 3.2 Important properties - Commutative: $a+b = b+a$, $ab = ba$ - Associative: $(a+b)+c = a+(b+c)$, $(ab)c = a(bc)$ - Distributive: $a(b+c) = ab+ac$ - Identity elements: additive identity $0$, multiplicative identity $1$ - Inverses: additive inverse $-a$, multiplicative inverse $a^{-1}$ for $a\neq 0$ ### 3.3 Even and odd integers - Even: divisible by $2$, written $2k$ for integer $k$ - Odd: one more than an even, written $2k+1$ ### 3.4 Prime and composite numbers > **Definition:** A prime number has exactly two positive divisors: $1$ and itself. A composite number has additional positive divisors. - Examples: primes $2,3,5,7,11$; composite $4,6,8,9$ - Fundamental Theorem of Arithmetic: every integer greater than $1$ can be written uniquely as a product of primes (up to order) ## 4. Fractions, Decimals and Percentages - Fraction $\frac{a}{b}$ represents division $a\div b$ - Convert fraction to decimal via division; repeating or terminating - Percentage is fraction times $100$: $\text{percent} = 100\times\frac{a}{b}$ Example: Convert $\frac{3}{8}$ to decimal and percent. $\frac{3}{8} = 0.375$, which is $37.5\%$. ## 5. Comparing and Approximating Numbers - Use numbe