Algebra and Trigonometry Introduction by Owen Borville April 23, 2025
Types of Numbers in Mathematics Including Real Numbers
Natural numbers are the set of positive whole numbers used for counting, such as 1, 2, 3, 4, 5, ...
Whole numbers are the set of positive natural numbers plus zero, such as 0, 1, 2, 3, ...
Integers are the set of numbers that add the opposites of natural numbers to the set of whole numbers (positive numbers, negative numbers, and zero) such as (..., -3, -2, -1, 0, 1, 2, 3, ...). Therefore, the set of positive integers are the natural numbers.
Rational numbers are the set of numbers (m/n) that can be expressed as the quotient or fraction of two integers, where m and n are integers and n is not equal to zero.
Rational numbers can also be expressed as a terminating decimal or repeating decimal. Terminating decimal = (1/8) = 0.125 Repeating decimal = (2/11) = 0.18181818 or 0.18 with a line over the number.
Irrational numbers cannot be expressed or written as fractions or as a fraction of two integers. A number is irrational if it is not rational. Irrational numbers do not terminate in decimal form and do not have a repeating pattern in decimal form. The number in decimal form continues forever with no pattern, such as the numbers pi and e, and the square roots of 2, 3, and 5.
Real numbers are the set of numbers that consist of both rational and irrational numbers. Any given number must be either a rational or irrational number, but not both. Therefore, real numbers include three groups: negative real numbers, zero, and positive real numbers. Negative and positive real numbers include fractions, decimals, and irrational numbers.
Real numbers can be shown on a horizontal number line known as the real number line with the zero in the middle, negative numbers to the left of zero, and positive numbers to the right of zero. Each number is spaced an equal, fixed distance apart. Each number designates a unique position on the number line and each position on the number line designates one real number.
Relationships between real number types can be shown in related subsets:
The first subset is the set of natural numbers. (1, 2, 3,...)
The next larger subset is the set of whole numbers. (0, 1, 2, 3,...)
The next larger subset is the set of integers. (..., -3, -2, -1, 0, 1, 2, 3,...)
The next larger subset is the set of rational numbers. (m/n, where m and n are integers and n is not zero)
The next subset is the set of irrational numbers. (numbers are not rational, non-repeating, and non-terminating, such as pi, e, and square roots of 2, 3, and 5)
Calculations and the Order of Operations
Exponential notation (a^n is the number a raised to the nth power), where a is the base and n is the exponent. An exponential notation can be part of a mathematical expression.
Mathematical expressions are evaluated using the order of operations, which is a sequence of rules for evaluating such expressions. The order of operations for mathematical expressions is as follows:
Order of Operations for mathematical expressions:
1 Parentheses are evaluated first. (also radicals or grouping symbols, absolute value)
2 Exponents are evaluated next.
3 Multiplication and Division expressions are evaluated next.
4 Addition and Subtraction expressions are evaluated lastly.
Properties of Real Numbers
Commutative Properties
The commutative property of addition says that numbers can be added in any order without affecting the value:
a+b=b+a
The commutative property of multiplication says that numbers can be multiplied in any order without affecting the value:
a*b = b*a
Neither subtraction or division is commutative.
Associative Properties
The associative property of multiplication says that numbers grouped in different ways give the same value when multiplied together.
a(bc)=(ab)c
The associative property of addition says that numbers may be grouped differently without affecting the sum.
a+(b+c) = (a+b)+c
Distributive Property
The distributive property says that the product of a factor times a sum is the sum of the factor times each term in the sum.
a*(b + c) = a*b + a*c
Identity Properties
The identity property of addition says that there is a unique number, the additive identity, that when added to a number the result is the original number.
a + 0 = a
The identity property of multiplication says that there is a unique number called the multiplicative identity that when multiplied by a number results in the original number.
a * 1 = a
Inverse Properties
The inverse property of addition says that for every real number a, there is a unique number, called the additive inverse or opposite, denoted by (-a) that when added to the original number results in the additive identity 0.
a + (-a) = 0
The inverse property of multiplication says that for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal) denoted (1/a) that when multiplied by the original number, results in the multiplicative identity, 1. The inverse property of multiplication applies for all real numbers except 0 because the reciprocal of 0 is not defined.
a * (1/a) = 1
Evaluating Algebraic Expressions
In algebraic expressions, a constant is a number that does not vary and a variable is a number that varies. An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
An equation is a mathematical statement indicating that two numerical or algebraic expressions are equal.
3x + 1 = 8
x = 7/3
A formula is an equation expressing a relationship between constant and variable quantities.
Simple interest = p * r * t = principal amount * interest rate * time (years)
Simplifying algebraic expressions can be done by adding or subtracting number expressions with the same variable and different coefficients.
4x + 2x = 6x
3y + 7y = 10y
2x + 3y + 4 + 3x = 5x + 3y + 4
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Exponents and Scientific Notation
Radicals and Rational Exponents
Polynomials
Factoring Polynomials
Rational Expressions
Types of Numbers in Mathematics Including Real Numbers
Natural numbers are the set of positive whole numbers used for counting, such as 1, 2, 3, 4, 5, ...
Whole numbers are the set of positive natural numbers plus zero, such as 0, 1, 2, 3, ...
Integers are the set of numbers that add the opposites of natural numbers to the set of whole numbers (positive numbers, negative numbers, and zero) such as (..., -3, -2, -1, 0, 1, 2, 3, ...). Therefore, the set of positive integers are the natural numbers.
Rational numbers are the set of numbers (m/n) that can be expressed as the quotient or fraction of two integers, where m and n are integers and n is not equal to zero.
Rational numbers can also be expressed as a terminating decimal or repeating decimal. Terminating decimal = (1/8) = 0.125 Repeating decimal = (2/11) = 0.18181818 or 0.18 with a line over the number.
Irrational numbers cannot be expressed or written as fractions or as a fraction of two integers. A number is irrational if it is not rational. Irrational numbers do not terminate in decimal form and do not have a repeating pattern in decimal form. The number in decimal form continues forever with no pattern, such as the numbers pi and e, and the square roots of 2, 3, and 5.
Real numbers are the set of numbers that consist of both rational and irrational numbers. Any given number must be either a rational or irrational number, but not both. Therefore, real numbers include three groups: negative real numbers, zero, and positive real numbers. Negative and positive real numbers include fractions, decimals, and irrational numbers.
Real numbers can be shown on a horizontal number line known as the real number line with the zero in the middle, negative numbers to the left of zero, and positive numbers to the right of zero. Each number is spaced an equal, fixed distance apart. Each number designates a unique position on the number line and each position on the number line designates one real number.
Relationships between real number types can be shown in related subsets:
The first subset is the set of natural numbers. (1, 2, 3,...)
The next larger subset is the set of whole numbers. (0, 1, 2, 3,...)
The next larger subset is the set of integers. (..., -3, -2, -1, 0, 1, 2, 3,...)
The next larger subset is the set of rational numbers. (m/n, where m and n are integers and n is not zero)
The next subset is the set of irrational numbers. (numbers are not rational, non-repeating, and non-terminating, such as pi, e, and square roots of 2, 3, and 5)
Calculations and the Order of Operations
Exponential notation (a^n is the number a raised to the nth power), where a is the base and n is the exponent. An exponential notation can be part of a mathematical expression.
Mathematical expressions are evaluated using the order of operations, which is a sequence of rules for evaluating such expressions. The order of operations for mathematical expressions is as follows:
Order of Operations for mathematical expressions:
1 Parentheses are evaluated first. (also radicals or grouping symbols, absolute value)
2 Exponents are evaluated next.
3 Multiplication and Division expressions are evaluated next.
4 Addition and Subtraction expressions are evaluated lastly.
Properties of Real Numbers
Commutative Properties
The commutative property of addition says that numbers can be added in any order without affecting the value:
a+b=b+a
The commutative property of multiplication says that numbers can be multiplied in any order without affecting the value:
a*b = b*a
Neither subtraction or division is commutative.
Associative Properties
The associative property of multiplication says that numbers grouped in different ways give the same value when multiplied together.
a(bc)=(ab)c
The associative property of addition says that numbers may be grouped differently without affecting the sum.
a+(b+c) = (a+b)+c
Distributive Property
The distributive property says that the product of a factor times a sum is the sum of the factor times each term in the sum.
a*(b + c) = a*b + a*c
Identity Properties
The identity property of addition says that there is a unique number, the additive identity, that when added to a number the result is the original number.
a + 0 = a
The identity property of multiplication says that there is a unique number called the multiplicative identity that when multiplied by a number results in the original number.
a * 1 = a
Inverse Properties
The inverse property of addition says that for every real number a, there is a unique number, called the additive inverse or opposite, denoted by (-a) that when added to the original number results in the additive identity 0.
a + (-a) = 0
The inverse property of multiplication says that for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal) denoted (1/a) that when multiplied by the original number, results in the multiplicative identity, 1. The inverse property of multiplication applies for all real numbers except 0 because the reciprocal of 0 is not defined.
a * (1/a) = 1
Evaluating Algebraic Expressions
In algebraic expressions, a constant is a number that does not vary and a variable is a number that varies. An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.
An equation is a mathematical statement indicating that two numerical or algebraic expressions are equal.
3x + 1 = 8
x = 7/3
A formula is an equation expressing a relationship between constant and variable quantities.
Simple interest = p * r * t = principal amount * interest rate * time (years)
Simplifying algebraic expressions can be done by adding or subtracting number expressions with the same variable and different coefficients.
4x + 2x = 6x
3y + 7y = 10y
2x + 3y + 4 + 3x = 5x + 3y + 4
-----------------------------------------------
Exponents and Scientific Notation
Radicals and Rational Exponents
Polynomials
Factoring Polynomials
Rational Expressions