MAT-172 Contemporary Math I
Upon completion the student will be able to:
- Explain the difference between inductive and deductive reasoning.
- Recognize inductive reasoning and illustrate with examples.
- Recognize deductive reasoning and illustrate with examples.
- Define a statement.
- Explain the advantage of using symbols to represent statements.
- Identify variations of negation.
- Construct a correct negation of a statement.
- Write compound statements using symbols.
- Translate symbolic statements into words.
- Create truth tables for compound statements.
- Identify and use converse, inverse, and contrapositive of a conditional.
- Determine the validity of an argument by using a truth table.
- Identify and give examples of each of the following:natural number, cardinal number, ordinal number, (positive or negative) integer, rational number, irrational number, real number, imaginary number, complex number
- Explain the structure of the real number system.
- Identify and give examples of terminating and repeating decimals.
- Identify and give examples of non-repeating decimals.
- Define a binary operation.
- Use tables to define binary operations.
- Define and verify commutative and associative properties for a binary operation.
- Define and verify the distributive property of one binary operation over a second.
- Define a mathematical system and give examples.
- Determine if an operation is closed.
- Define and find the identify element for a binary operation.
- Define and find the inverse element for a binary operation.
- Define a group and give examples.
- Determine if a set and operation form a group.
- Define and identify an Abelian (commutative) group.
- Define a field and give examples.
- Define a ring and give examples.
- Determine if a set and two operation form a field.
- Define modular (clock) arithmetic.
- Determine congruence classes modulo a given integer.
- Construct tables for modular arithmetic.
- Solve equations in modulo systems, using proper notation.
- Explain the relationship between modular arithmetic and groups and fields.
- Compare and contrast grouping, placement, positional, and place value systems of numeration.
- Recognize the historical and cultural significance of a zero symbol.
- Write and evaluate expanded notation for base n numerals.
- Convert base ten numerals to base n numerals and vice versa.
- Perform arithmetic operations in base n, especially base two (binary), eight (octal), twelve, sixteen (hexadecimal), and sixty (sexagesimal) numeration systems.
- Use properties of numeration systems for recreational mathematics.
- Determine the divisibility of a number by 2, 3, 4, 5, 6, 9, and 10.
- Define a prime number.
- Define a composite number.
- Use the Sieve of Eratosthenes to find prime numbers.
- Find the prime factorization of a number.
- Find the greatest common factor of two or more numbers.
- Find the least common multiple of two or more numbers.
- State and explain Fermat’s last Theorem.
- Explain and illustrate Goldbach’s Conjectures.
- Define a Mersenne prime number.
- Identify and solve linear Diophantine equations.
- Illustrate and explain non-linear Diophantine equations.
SHAPES, PATTERNS AND SYMMETRY
- Identify and characterize arithmetic sequences.
- Identify and characterize geometric sequences
- Identify and give examples of Fibonacci sequence in nature.
- Define the Golden Ratio.
- Detect patterns of various kinds in music, art, and surroundings.
- Explain and give examples of various rigid motions:reflection, rotation, translation, and glide reflection.
- Identify symmetries associated with rigid motions.
- Be familiar with the notion of a fractal, its construction, and uses in modeling.
- Understand scaling factors and their effect of measures of area, volume, and surface area of 2- and 3-dimensional figures.
- Identify the five regular polyhedral solids.