Basics of Mathematics

Mathematics Basics

• Mathematics has its roots in the Greek words "manthanein," meaning learning, and "techne," referring to technique.
• Essentially, mathematics is the art of learning concerning various disciplines.
• Mathematics is defined as the science of quantity, measurement, and special relations.
• Archimedes is considered the father of mathematics.

Applications of Math in Daily Life

• Cooking demands accurate measurement of ingredients (tablespoons, teaspoons, ounces, grams, kilograms) to avoid disastrous results.
• Overweight individuals need to monitor their calorie intake by using math to calculate the total calories consumed daily, especially on special weight loss diets.
• The field of banking relies heavily on math, as basic mathematical knowledge is essential for banking operations.
• Artists use math for paintings, designing collages, and during dance performances to determine sizes and proportions accurately.
• In medicine, medical prescriptions involve precise calculations.
• X-rays and scans employ mathematical concepts, originally presented as graph plots.
• Healthcare workers need math skills for various tasks.
• Essential math skills are a must for healthcare professionals.
• Mistakes in healthcare math can have life-threatening consequences.

Basic Mathematical Symbols

• Includes equals, various arithmetic symbols, inequality symbols, and more
• Exponents, roots, modulo, percentages etc.

Branches of Mathematics

• Applied Mathematics
• Linear Algebra
• Differential Equations
• Number Theory
• Game Theory
• Cryptography
• Real Analysis
• Trigonometry
• Number System
• Geometry
• Topology
• Statistics
• Symmetry
• Complex Analysis
• Analysis
• Group Theory
• Calculus
• Set Theory
• Algebra
• Logic
• Numerical Analysis
• Probability

Algebra

• Algebra uses letters to represent numbers.
• Muhammad Bin Musa Al-Khwarizmi is the father of algebra.

Trigonometry

• Trigonometry is a field that deals with right triangles.
• The shape of a right triangle results from the value of either of the other two angles.
• Once one angle is known, the ratios of the various sides in the triangle remain the same across sizes.
• These ratios get defined by trigonometric functions based on the known angle.
• If one angle and one side length of a right triangle are known, all other angles and side lengths can be calculated.
• Alternatively, knowing two sides of the triangle means that the third side and all angles can be determined.
• Trigonometric functions relate an angle in a right-angled triangle to the ratio of two of its sides.
• Sine, Cosine and Tangent are trigonometric functions.
• The angle in question is often called theta (θ).
  • sin(θ) = Opposite / Hypotenuse
  • cos(θ) = Adjacent / Hypotenuse
  • tan(θ) = Opposite / Adjacent

Geometry

• Geometry deals with the properties and relationships of points, lines, angles, curves, surfaces, and solids,
• Geometry is the visual study of shapes, sizes, patterns, and positions.
• Euclid is considered the "Father of Geometry."
• Euclid's contribution resulted in the geometry referred to today as Euclidean geometry.

Lines, Rays, and Angles

• Lines have neither a beginning nor an end and continue indefinitely in both directions.
• Line segments have a distinct beginning and endpoint.
• Line segments create a shape's sides.
• Rays start at a point but do not have an endpoint.
• Parallel lines exist in the same plane without intersecting.
• Intersecting lines meet at a shared point.
• Perpendicular lines intersect and form a right or 90-degree angle.
• Angles form two rays with the same starting point (vertex).
• The two rays that create the angle are called its sides.

Measuring Angles

• Angles can be measured using a protractor
• Acute angles are less than 90 degrees.
• Right angles are exactly 90 degrees.
• Obtuse angles are greater than 90 degrees, and less than 180 degrees.
• Straight angles are exactly 180 degrees.
• Reflex angles measure greater than 180 degrees.
• Full rotation angles measure exactly 360 degrees.

Calculus

• Calculus involves the study of continuous change and has two primary branches.
• The 2 branches of calculus are differential calculus and integral calculus.
• Examples of calculus include how bank balances increase or decrease and how data accumulates.
  • Derivative: Deals with the rate at which something changes.
  • Integral: Deals with how much something accumulates.
• Requires knowledge of precalculus (algebra, geometry, and trigonometry).
• Isacc Newton developed Calculus.
• Calculus determines the best drug dosage administered and its rate of dissolving in medicine.

Forensic Science

• Calculus determines victim location and instrument in forensic science.
• Calculus serves as a DNA reservoir when destructive analysis of teeth/bones is off limits.
• Calipers determine length/width to determine the angle of impact of the blood drop using the law of sine.
• Height determines the blood amount from the law of tangents.
• Forensic math shows forensics students use math, science, and technology to solve crimes

Blood Spatter Analysis

• In blood spatter analysis, impact angles determine each drop of blood.
• Measure length and width of each blood droplet.
• Assess the R value using the formula R = width/length for each drop
• Once you determine the R value, use the sine table or calculator
• The determination of the impact angle use the arcsin of width/length
• Bloodstain with a 0.08-inch width and 0.16-inch length has a width/length of 0.5.
• The arcsin of (inverse sin) 0.5 yields 30 degrees, meaning it hit the surface at 30 degrees Bloodstain with a 0.04-inch width and 0.16-inch length has an impact angle of about 14.5 degrees.
• The area of covergence is determined by drawing lines