Review of Basic Number Theory Concepts
COURSE OUTCOMES
Create visualizations and solutions for some problems involving basic number theory concepts, basic algebraic concepts, equations, inequalities, systems of linear equations, and matrices with the use of computer applications.
Review of Basic Number Theory Concepts
LEARNING OUTCOMES
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Identify some basic number theory concepts, real number system, and complex number system
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Make solutions and illustrations for some problems involving fundamental number theory concepts, real number system, and complex number system
SET
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are used in real life to organize and categorize things that share a common property or attribute.
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a set, in mathematics, is a collection of distinct objects or elements that share a common property or properties. These objects can be anything, including numbers, letters, or even other sets.
Variety of set of clothes as an example.
CLOSET
SET FORMS
DESCRIPTIVE FORM
A set is described using natural language
The set of all shapes with four sides
SET-BUILDER FORM
A set is defined by a rule or condition
{S | S is a shape with four sides}
ROSTER FORM
A set is defined by listing all of its elements
{square, rectangle, parallelogram, trapezoid, rhombus}
REAL NUMBER SYSTEM
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is a set of numbers that includes all the rational and irrational numbers. In other words, it includes all the numbers that can be represented as decimals or fractions. The real number system is denoted by the symbol R.
NATURAL NUMBERS (ℕ)
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are a set of numbers that includes all the positive integers, starting from 1 and going to infinity.
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They include all positive integers, but not zero or negative numbers. Examples of natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and so on.
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are used in many different areas of mathematics, as well as in science, engineering, and other fields where counting and measuring are important.
WHOLE NUMBERS (W)
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are simply the set of numbers that includes all the natural numbers (positive integers) and zero.
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start from zero and continue with counting numbers, such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and so on.
INTEGERS (ℤ)
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are a set of numbers that includes all the natural numbers (positive integers), their additive inverses (negative integers), and zero. Integers are denoted by the symbol Z.
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Examples of integers are -3, -2, -1, 0, 1, 2, 3, and so on.
PROPERTIES
Closure Property
the result will always be a number within that same set.
3 + 4 = 7
Distributive Property
you can distribute the multiplication over each number and then add (or subtract) the results.
2(3 + 4) is the same as 2 x 3 + 2 x 4
Identity Property
when used with that operation, leaves any other element unchanged. For addition, the identity element is zero, and for multiplication, the identity element is one.
6 x 1 = 6
Commutative Property
order of the operands does not affect the outcome of the operation.
2 + 3 = 5 and 3 + 2 = 5.
Additive Inverse Property
when added to the original number, gives a sum of zero. In other words, the sum of a number and its additive inverse is zero.
7 + (-7) = 0
Associative Property
grouping of operands does not affect the outcome of the operation.
(2 + 3) + 4 is the same as 2 + (3 + 4)
Multiplicative Inverse Property
when multiplied by the original number, gives a product of one. In other words, the product of a number and its multiplicative inverse is one.
1/3 x 3 = 1
RATIONAL NUMBERS (ℚ)
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are numbers that can be expressed as a ratio of two integers (where the denominator is not zero). In other words, a rational number can be written in the form p/q, where p and q are integers and q is not equal to zero.
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1/2 is a rational number because it can be written as the ratio of two integers, 1 and 2.
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2/3 is a rational number because it can be written as the ratio of two integers, 2 and 3.
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-3/4 is a rational number because it can be written as the ratio of two integers, -3 and 4.
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0.6 is a rational number because it can be written as the ratio of two integers, 3 and 5 (0.6 = 3/5).
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-1.25 is a rational number because it can be written as the ratio of two integers, -5 and 4 (-1.25 = -5/4).
IRRATIONAL NUMBERS (ℚ)
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are numbers that cannot be expressed as a ratio of two integers. In other words, they cannot be written in the form p/q, where p and q are integers and q is not equal to zero. Irrational numbers include non-repeating, non-terminating decimals.
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√2 is an irrational number because it cannot be expressed as the ratio of two integers. Its decimal representation goes on infinitely without repeating: 1.41421356...
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π (pi) is an irrational number because it cannot be expressed as the ratio of two integers. Its decimal representation goes on infinitely without repeating: 3.14159265...
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√3 is an irrational number because it cannot be expressed as the ratio of two integers. Its decimal representation goes on infinitely without repeating: 1.73205080...
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e (Euler's number) is an irrational number because it cannot be expressed as the ratio of two integers. Its decimal representation goes on infinitely without repeating: 2.71828182...
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√5 is an irrational number because it cannot be expressed as the ratio of two integers. Its decimal representation goes on infinitely without repeating: 2.23606798...
REAL NUMBERS (ℝ)
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Real numbers are the set of all rational and irrational numbers. They include any number that can be represented on the number line.
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5 is a real number because it can be represented on the number line.
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-2.5 is a real number because it can be represented on the number line.
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√2 is a real number because it is an irrational number and thus a subset of real numbers.
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3/4 is a real number because it is a rational number and thus a subset of real numbers.
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π (pi) is a real number because it is an irrational number and thus a subset of real numbers.
OTHER NUMBERS
ABUNDANT NUMBER
A number is said to be abundant if the sum of its proper divisors (excluding itself) is greater than the number itself.
12 is an abundant number because its proper divisors (excluding itself) are 1, 2, 3, 4, and 6, which add up to 16, which is greater than 12.
Distributive Property
Two numbers are said to be amicable if each is the sum of the proper divisors of the other.
The smallest pair of amicable numbers is (220, 284). The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, which add up to 220.
DEFICIENT NUMBER
A number is said to be deficient if the sum of its proper divisors is less than the number itself.
15 is a deficient number because its proper divisors are 1, 3, and 5, which add up to 9, which is less than 15.
PERFECT NUMBER
A number is said to be perfect if the sum of its proper divisors is equal to the number itself.
6 is a perfect number because its proper divisors are 1, 2, and 3, which add up to 6.
