MATH 1ST SEM

Mathema (Ancient Greek)

Meant "that which is learned" or "a lesson"; derived from manthanein.

Manthanein

Means "to learn."

Pythagoreans

believed mathematics was the "true learning." 6th century BCE;

Mathematica (Latin)

Roman form of mathema; during the Middle Ages it referred to astrology or fortune–telling because stars were studied using math.

16th Century English

Mathematics became the standard term, regaining its focus on geometry and arithmetic.

Math Literal Meaning

"The Act of Learning."

History of Mathematics

Journey from simple counting for survival, the "universal language."

Ancient Foundations (3000–500 BCE)

Mathematics developed from trade, taxation, and construction.

Greek Revolution (600 BCE–400 CE)

Shifted mathematics from calculation to logical proof.

Renaissance & Birth of Calculus (1500–1700)

Mathematics advanced rapidly as knowledge returned to Europe.

Golden Age of India & Islamic World (400–1200 CE)

Mathematics flourished in the East.

Modern & Abstract Era (1800–Present)

Mathematics became more abstract, focusing on logic and the foundations of the universe.

Sumerians & Babylonians

Developed the base–60 (sexagesimal) system, giving us 60 seconds per minute and 360 in a circle; understood the Pythagorean Theorem before Pythagoras.

Egyptians

Used geometry and arithmetic for pyramid construction and land surveying; used a decimal system and fractions.

Thales & Pythagoras

Introduced mathematical proof.

Archimedes

Anticipated modern calculus using infinitesimals to calculate area and volume.

Brahmagupta (India)

Developed zero as a number and the decimal place–value system.

Modern Mathematics

Foundation of cryptography, artificial intelligence, and quantum physics.

Mathematics

Study of patterns, relationships, and order; a universal language, tool for reasoning, and way of thinking. Literally means "The Act of Learning."

Precision

Uses exact definitions and clear rules; no guesswork.

Logical Order

Every conclusion follows step by step reasoning.

Universality

The same mathematical rules apply everywhere.

Pattern–Centered

Finds order in numbers, events, and nature.

Partly Invented

As old as humanity; began as a way to represent quantity, form, pattern, and change.

Mathematics Everyday Life

Used in personal life, community, technology & services, and nature

Nature of Mathematics

A system of knowledge and relationships discovered through accepted rules of reasoning.

Pythagorean Theorem

Example of a mathematical truth discovered rather than invented.

Ishango Bone

Baboon fibula found near the Semliki River with notches believed to track lunar phases in relation to women's menstrual cycles.

Sexagesimal System

Base–60 system developed by the Sumerians; used for time, irrigation, land division, taxation, counting livestock and grains, and tracking celestial bodies.

Modern Binary System

Base–2 system using 0 and 1; foundation of modern computers.

Alan Turing & Kurt Godel

Explored the limits of computation and mathematical proof, leading to modern computing.

Gottfried Leibniz (1679)

Introduced the modern binary system.

Leibniz & Isaac Newton

Independently developed early Calculus.

Euclid

"Father of Geometry"; wrote The Elements, the standard math textbook for over 2,000 years.

Al–Khwarizmi

Wrote the first systematic book on algebra (al–jabr); origin of the word "algorithm."

René Descartes

Created Analytic Geometry (Cartesian coordinate plane) by linking algebra and geometry.

Newton & Leibniz

Independently developed Calculus to describe change and motion.

Gauss

"Prince of Mathematicians"; revolutionized number theory.

Galois

Founded group theory.

Much of Mathematics is Discovered

Pythagorean Theorem; mathematical foundation invented, later developments discovered through reasoning.

Patterns

Regular, repeated, or recurring forms/designs (Aufmann et al., 2018).

Patterns in Nature

Frequently observed in nature and the world.

Symmetry

Butterfly wings, flower petals, mango leaves, snowflakes; one half mirrors the other; balance & efficiency.

Fibonacci Pattern

Sunflower seeds, pinecone scales, pineapple skin, snail shells; new part grows based on previous two; maximizes seeds/leaves in small space.

Tessellations

Honeycomb cells, fish scales, tiled floors; shapes fit without gaps/overlapping; hexagons hold most honey with least wax.

Periodic Patterns

Day/night, high/low tide, seasons, heartbeat; repeat at fixed intervals; predictable regular cycle.

Spirals

Galaxy shapes, typhoon cloud bands, leaf arrangement; grow outward evenly without blocking sunlight/space.

Sequence

List of numbers/shapes following a specific rule.

Arithmetic Sequence

Add/subtract same number each time.

Geometric Sequence

Multiply/divide by same number each time.

Fibonacci Sequence

Add two preceding numbers to get next.

Mixed Pattern

Alternating rules; +4, −2, repeat.

Language

Systematic communication of ideas/feelings using conventionalized signs, sounds, gestures, or marks with understood meanings.

Mathematical Language

System for communicating mathematical ideas, natural language + technical terms/grammatical conventions + specialized symbolic notation for formulas.

Non–temporal

Independent of or unaffected by time.

No Emotional Content

No strong feelings expressed.

Mathematical_Language

Precise & concise.

+

Addition

−

subtraction

±

positive/negative signs

×

Multiplication/ also represented by dot

÷

Division/ represented by backslash or horizontal bar

=

Equality sign.

≠

Not equal.

≈

Approximately equal.

≡

Congruent/equivalent.

>

Greater than.

<

Less than.

≥

Greater than or equal to.

≤

Less than or equal to.

x, y, z

Lowercase letters represent variables/placeholders for unknown values.

P, V, T

Uppercase letters used in physics; pressure, volume, temperature.

√x

Nonnegative number whose square equals x.

ⁿ√x

nth root of x.

t₁, t₂, t₃…

First, second, third, etc. observed values of variable t.

α, β, γ

Greek letters; alpha, beta, gamma; used to represent angles.

Σ

Sigma; summation.

μ

Mu; population mean.

∫

Integral sign; integration.

dy/dx

Rate of change.