1.1 Introduction to Whole Numbers

Whole numbers, also known as natural numbers, are the set of positive integers that begin with 1 and continue indefinitely. They are the most fundamental and commonly used numbers in mathematics, with wide applications in various fields.

Integers: Integers are the set of whole numbers, including negative numbers and zero.

Prime Numbers: Prime numbers are whole numbers greater than 1 that have no positive divisors other than 1 and themselves.

Counting Numbers: Counting numbers, or natural numbers, are the positive whole numbers starting from 1 and continuing indefinitely.

Zero is a fundamental numerical concept that represents the absence of quantity or magnitude. It serves as a starting point and a reference for various mathematical operations and number systems, making it a crucial element in the understanding of whole numbers, integers, and the properties of identity, inverses, and zero.

Additive Identity: The property that any number added to zero results in the original number, with zero acting as the additive identity element.

Multiplicative Identity: The property that any number multiplied by one results in the original number, with one acting as the multiplicative identity element.

Additive Inverse: The property that a number and its additive inverse (opposite) sum to zero, with zero acting as the additive identity.

Place value is a fundamental concept in mathematics that describes the value of a digit based on its position within a number. It is the foundation for understanding and working with whole numbers, decimals, and other numerical representations.

Digit: A single symbol used to represent a number, such as 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Numeral System: A system for representing numbers, such as the decimal (base 10) system or the binary (base 2) system.

Expanded Notation: A way of writing a number that shows the value of each digit based on its place value.

Rounding is the process of approximating a numerical value to a simpler value, typically to a specified number of decimal places or significant figures. It is a fundamental concept in mathematics and is particularly relevant in the context of whole numbers and decimals.

Significant Figures: Significant figures refer to the meaningful digits in a number, including all non-zero digits and any zeros that are between non-zero digits or at the end of the number.

Place Value: Place value is the value of a digit in a number based on its position, such as ones, tens, hundreds, etc. for whole numbers, and tenths, hundredths, thousandths, etc. for decimals.

Estimation: Estimation is the process of approximating a value, often by rounding, to get a reasonable idea of the quantity without calculating an exact answer.

Natural numbers, also known as counting numbers, are the set of positive integers starting from 1 and continuing without end. They are the most basic and fundamental numbers used in mathematics, forming the foundation for numerical operations and quantification.

Whole Numbers: Whole numbers include the natural numbers as well as the number zero, representing a complete set of counting numbers.

Integers: Integers expand the set of natural numbers to include negative numbers, zero, and positive numbers, creating a more comprehensive numerical system.

Rational Numbers: Rational numbers are numbers that can be expressed as a ratio of two integers, including natural numbers, whole numbers, and fractions.

The ones place in a whole number represents the individual units, or ones, of that number. The ones digit is the rightmost digit in a whole number and indicates the quantity of single units present.

Digit: A single symbol used to represent a number, such as 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Place Value: The value of a digit based on its position within a number, such as ones, tens, hundreds, etc.

Whole Numbers: The set of positive integers, including 0, that have no fractional part.

Tens refers to the place value position in the base-10 numeral system that represents groups of ten. It is the second-most significant digit, with each digit in the tens place representing a value ten times greater than the digit in the ones place.

Place Value: The numerical value assigned to a digit based on its position in a number, with each successive place representing a value ten times greater than the previous place.

Base-10 Numeral System: The most commonly used number system, which uses ten unique digits (0-9) and assigns place values that increase by a factor of ten from right to left.

Ones: The place value position that represents individual units, with each digit in the ones place having a value of 1, 2, 3, etc.

Hundreds refers to the place value position in the base-10 number system that represents amounts of one hundred or multiples of one hundred. It is a key concept in understanding whole numbers and performing addition operations.

Place Value: The value of a digit in a number based on its position, such as ones, tens, hundreds, thousands, and so on.

Base-10 Number System: The numbering system that uses 10 digits (0-9) and where each digit's value is determined by its place value.

Whole Numbers: The set of positive integers, including 0, that can be represented without a fractional component.

Thousands is a unit of measurement used to denote a quantity of an object or value that is one thousand times greater than a single unit. It is a way of expressing large numbers in a more concise and manageable form.

Place Value: The value of a digit in a number based on its position, such as ones, tens, hundreds, and thousands.

Numeral System: A system for representing numbers, such as the decimal system which uses the digits 0-9 and place value to represent any number.

Rounding: The process of approximating a number to a specified place value, such as rounding to the nearest thousand.

Standard form is a way of expressing numbers, equations, or other mathematical entities in a specific, organized, and easily recognizable format. It provides a consistent and concise way to represent these elements, making them easier to work with, compare, and manipulate across various mathematical contexts.

Expanded Form: A way of writing a number by listing the value of each digit in its place value, such as 4,567 = 4,000 + 500 + 60 + 7.

Scientific Notation: A way of expressing very large or very small numbers using a decimal number multiplied by a power of 10, such as 4.567 x 10^3 for 4,567.

Polynomial Form: The representation of a polynomial expression in a standard format, with terms arranged in descending order of the exponents of the variable(s).

Expanded form is a way of representing a number by breaking it down into its place value components. It demonstrates the value of each digit in a number by showing the value of each place, such as ones, tens, hundreds, and so on.

Place Value: The value of a digit based on its position in a number, such as ones, tens, hundreds, and so on.

Standard Form: The conventional way of writing a number, where all the digits are written together without spaces or commas.

Word Form: The representation of a number using words, such as 'one hundred and twenty-three.'

Word form refers to the representation of a number using words rather than numerals. It is a way of expressing quantities and values in a written, non-numerical format. This concept is important in both the understanding of whole numbers and the comprehension of decimal values.

Numeral: A symbol or figure used to represent a number, such as the digits 0-9.

Expanded Form: A way of writing a number that shows the value of each digit, such as 123 being written as 100 + 20 + 3.

Place Value: The value of a digit in a number based on its position, such as the ones, tens, hundreds, etc.

The less than symbol (<) is a mathematical symbol used to indicate that one value is smaller or less than another value. It is used to compare quantities and express relationships between numbers or other mathematical entities.

Greater Than (>): The greater than symbol is the opposite of the less than symbol, indicating that one value is larger or greater than another value.

Inequality: An inequality is a mathematical statement that shows a relationship between two expressions where one is less than, greater than, or not equal to the other.

Order of Operations: The order of operations is a set of rules that determine the sequence in which operations should be performed in a mathematical expression, including the use of the less than symbol.

The greater than symbol (>) is a mathematical operator used to compare two values and indicate that one value is larger or greater than the other. It is a fundamental concept in the introduction to whole numbers, as it allows for the ordering and comparison of different whole numbers.

Whole Numbers: Whole numbers are the set of positive integers, including 0, that can be represented without a fractional component (e.g., 0, 1, 2, 3, 4, etc.).

Inequality: An inequality is a mathematical statement that compares two values using symbols such as >, <, ≥, or ≤ to indicate the relationship between them.

Order of Operations: The order of operations is a set of rules that determines the sequence in which mathematical operations should be performed to evaluate an expression correctly.

Addition is a fundamental mathematical operation that combines two or more numbers or quantities to find their sum. It is a way of combining values to obtain a total or overall amount. This key term is essential in understanding various mathematical concepts and operations within the context of pre-algebra.

Addend: The numbers or quantities that are being added together.

Sum: The result or total obtained by adding two or more numbers or quantities.

Commutative Property: The property that states that the order of the addends does not affect the sum.

Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is one of the four basic arithmetic operations, along with addition, subtraction, and division. Multiplication is used to find the total number of items or the area of a rectangle, and it is a fundamental concept in various mathematical contexts, including algebra, geometry, and statistics.

Factor: A factor is a number that divides another number evenly, without a remainder. Factors are used in the process of multiplication to determine the product of two or more numbers.

Commutative Property: The commutative property of multiplication states that the order of the factors does not affect the product. For example, 3 × 4 = 4 × 3.

Distributive Property: The distributive property of multiplication states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For example, 3 × (4 + 5) = (3 × 4) + (3 × 5).

Division is a fundamental mathematical operation that involves partitioning a quantity into equal parts or groups. It represents the inverse of multiplication, allowing us to find how many times one number is contained within another. This key term is essential in understanding various mathematical concepts, from whole numbers to exponents and scientific notation.

Dividend: The number that is being divided in a division operation.

Divisor: The number by which the dividend is being divided.

Quotient: The result of a division operation, representing the number of times the divisor goes into the dividend.

Odd numbers are integers that are not divisible by 2 without a remainder. They are the set of numbers that include 1, 3, 5, 7, 9, and so on, and are always separated by 2 from the next even number in the number line.

Even Numbers: Even numbers are integers that are divisible by 2 without a remainder. They include 2, 4, 6, 8, 10, and so on.

Prime Numbers: Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and themselves. Many odd numbers are prime, such as 3, 5, 7, and 11.

Composite Numbers: Composite numbers are positive integers that have at least one positive divisor other than 1 or the number itself. Odd composite numbers include 9, 15, and 21.

Even numbers are integers that are divisible by 2 without a remainder. They are a subset of whole numbers and are characterized by their ability to be evenly divided into two equal parts.

Odd Numbers: Odd numbers are integers that are not divisible by 2 without a remainder. They are the numbers that cannot be evenly divided into two equal parts.

Prime Numbers: Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and themselves.

Composite Numbers: Composite numbers are positive integers that have at least one positive divisor other than 1 or the number itself.