# MAT-172 Objectives

## MAT-172 Contemporary Math I

### LOGIC

1. Explain the difference between inductive and deductive reasoning.
2. Recognize inductive reasoning and illustrate with examples.
3. Recognize deductive reasoning and illustrate with examples.
4. Define a statement.
5. Explain the advantage of using symbols to represent statements.
6. Identify variations of negation.
7. Construct a correct negation of a statement.
8. Write compound statements using symbols.
9. Translate symbolic statements into words.
10. Create truth tables for compound statements.
11. Identify and use converse, inverse, and contrapositive of a conditional.
12. Determine the validity of an argument by using a truth table.

### MATHEMATICAL SYSTEMS

1. 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
2. Explain the structure of the real number system.
3. Identify and give examples of terminating and repeating decimals.
4. Identify and give examples of non-repeating decimals.
5. Define a binary operation.
6. Use tables to define binary operations.
7. Define and verify commutative and associative properties for a binary operation.
8. Define and verify the distributive property of one binary operation over a second.
9. Define a mathematical system and give examples.
10. Determine if an operation is closed.
11. Define and find the identify element for a binary operation.
12. Define and find the inverse element for a binary operation.
13. Define a group and give examples.
14. Determine if a set and operation form a group.
15. Define and identify an Abelian (commutative) group.
16. Define a field and give examples.
17. Define a ring and give examples.
18. Determine if a set and two operation form a field.
19. Define modular (clock) arithmetic.
20. Determine congruence classes modulo a given integer.
21. Construct tables for modular arithmetic.
22. Solve equations in modulo systems, using proper notation.
23. Explain the relationship between modular arithmetic and groups and fields.
24. Compare and contrast grouping, placement, positional, and place value systems of numeration.
25. Recognize the historical and cultural significance of a zero symbol.
26. Write and evaluate expanded notation for base n numerals.
27. Convert base ten numerals to base n numerals and vice versa.
28. Perform arithmetic operations in base n, especially base two (binary), eight (octal), twelve, sixteen (hexadecimal), and sixty (sexagesimal) numeration systems.
29. Use properties of numeration systems for recreational mathematics.

### NUMBER THEORY

1. Determine the divisibility of a number by 2, 3, 4, 5, 6, 9, and 10.
2. Define a prime number.
3. Define a composite number.
4. Use the Sieve of Eratosthenes to find prime numbers.
5. Find the prime factorization of a number.
6. Find the greatest common factor of two or more numbers.
7. Find the least common multiple of two or more numbers.
8. State and explain Fermat’s last Theorem.
9. Explain and illustrate Goldbach’s Conjectures.
10. Define a Mersenne prime number.
11. Identify and solve linear Diophantine equations.
12. Illustrate and explain non-linear Diophantine equations.

### SHAPES, PATTERNS AND SYMMETRY

1. Identify and characterize arithmetic sequences.
2. Identify and characterize geometric sequences
3. Identify and give examples of Fibonacci sequence in nature.
4. Define the Golden Ratio.
5. Detect patterns of various kinds in music, art, and surroundings.
6. Explain and give examples of various rigid motions:reflection, rotation, translation, and glide reflection.
7. Identify symmetries associated with rigid motions.
8. Be familiar with the notion of a fractal, its construction, and uses in modeling.
9. Understand scaling factors and their effect of measures of area, volume, and surface area of 2- and 3-dimensional figures.
10. Identify the five regular polyhedral solids.