MATH1525: Basic Mathematics for Science and Technology
Build the algebraic and geometric foundations for science and technology: functions, equations, trigonometry, vectors, matrices, sequences, inequalities, polar coordinates and logarithms, with worked examples and complete practice solutions.
Functions, Equations, and Scientific Models
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Chapter 1
Functions, Equations, and Scientific Models
A function includes a domain, not merely a formula. Domain restrictions survive simplification and composition. State the assumptions that make an equation transformation reversible. Check candidates after squaring.
- Read section
Functions, Equations, and Scientific Models
Control domains, reversible equations, inverse functions and scientifically admissible models.
Chapter 2
Trigonometry, Vectors, and Scientific Coordinates
Radians connect arc length with angular coordinates. Trigonometric inverses require restricted branches, while trigonometric equations can have many periodic solutions.
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Trigonometry, Vectors, and Scientific Coordinates
Use radians, inverse branches, periodic models and rectangular components, with an optional vector bridge.
Chapter 3
Polynomial Structure and Coordinate Geometry
Factors reveal roots and sign changes, while repeated roots require separate zero cases. Completing squares derives quadratic roots, extrema and geometric forms.
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Polynomial Structure and Coordinate Geometry
Connect polynomial structure, inequalities and rectangular loci, with optional conic enrichment.
Chapter 4
Linear Balances and Matrix Preparation
Coordinates turn lines and planes into scalar equations. Matrices collect simultaneous balances, and reversible elimination preserves all solutions.
- Read section
Linear Balances and Matrix Preparation
Organize simple matrices and solve elementary balance systems; spatial geometry is optional enrichment.
Chapter 5
Summation, Products, and Finite Structure
A finite sum lists additions; a finite product lists multiplications. Endpoints and the number of terms are part of the meaning. Sums are linear; products distribute over multiplication, not over addition.
- Read section
Summation, Products, and Finite Structure
Read, transform and prove finite sum and product formulas, including data summaries and empty ranges.
Chapter 6
Infinite Sequences, Series, and Error
A sequence converges when its entire tail lies within every prescribed tolerance of one finite number. An infinite series converges when the sequence of its finite partial sums does so.
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Infinite Sequences, Series, and Error
Distinguish sequences from series and justify geometric, telescoping and alternating approximations.
Chapter 7
Identities, Inequalities, and Mathematical Induction
An identity needs a domain; an equation needs a solution set. Reversible steps prove equivalence only where their operations are permitted.
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Identities, Inequalities, and Mathematical Induction
Verify identities and inequalities with domains and equality cases, and prove integer statements by induction.
Chapter 8
Polar Coordinates and Complex Numbers
Polar conversion requires a radius convention and a quadrant-aware angle. Multiple coordinate pairs can represent one point, especially at the origin.
- Read section
Polar Coordinates and Complex Numbers
Convert polar coordinates and develop complex arithmetic, modulus, arguments, powers, roots and loci.
- Read section
G3 Full Graphical Practice: Polar Pairs, Circle and Origin
Optional G3 full graphical practice: interpret polar coordinate pairs, construct a circle, and check the origin with complete worked solutions.
Chapter 9
Powers, Logarithms, and Multiplicative Scale
Integer powers are repeated multiplication and reciprocals. Rational powers require root and sign conventions; arbitrary real-exponent laws are safe with positive bases.
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Powers, Logarithms, and Multiplicative Scale
Apply safe power and logarithm laws to equations, inequalities, integer error thresholds and multiplicative scales.