University of Kwazulu-Natal

Real Numbers Algebra 101

Bonisiwe Shabane
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real numbers algebra 101

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization. Three to four thousand years ago, Egyptians introduced fractions.

They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts. But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further. Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions. Because of the evolution of the number system, we can now perform complex calculations using several categories of real numbers.

In this section we will explore sets of numbers, perform calculations with different kinds of numbers, and begin to learn about the use of numbers in algebraic expressions. The numbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as {1, 2, 3, …} where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3,…}.

The set of integers adds the opposites of the natural numbers to the set of whole numbers: {…,-3, -2, -1, 0, 1, 2, 3,…}. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers. The set of rational numbers is written as [latex]\left\{\frac{m}{n}|m\text{ and }{n}\text{ are integers and }{n}\ne{ 0 }\right\}[/latex]. Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0.

We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1. Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either: Sarah Lee AI generated o3-mini 9 min read · May 19, 2025 Real numbers form the bedrock of algebraic thinking. They are not just abstract symbols; rather, they represent quantities in the physical world—from distances and weights to temperatures and probabilities.

This article presents an in-depth guide to understanding real numbers, their classifications, operations, properties, and numerous applications in college algebra. Whether you're a student or an educator, mastering the fundamentals of real numbers is crucial for solving complex mathematical problems and modeling real-world situations. Definition of Real Numbers: Real numbers encompass all numbers that can be found on the number line. This includes both rational numbers (numbers that can be expressed as a fraction of two integers) and irrational numbers (numbers that cannot be expressed as a simple fraction). Formally, a real number \( x \) is an element of the set \( \mathbb{R} \). Importance in Algebra: In algebra, real numbers serve as constants, coefficients, and solutions to equations.

They are essential for: To illustrate these ideas visually, consider the following Mermaid diagram, which maps out the basic structure of real numbers: Which line plot corresponds to the inequality below? A closed circle indicates "greater than or equal to" or "less than or equal to," while and open circle indicates "greater than" or "less than". We can tell from the inequality that our line plot will only have closed circles, as the only symbol is "greater than or equal to." Our plot will show a closed circle on , and extend infinitvely in the positive direction.

What inequality is represented by the number line? An open circle indictes "less than" or "greater than," while a closed circle indicates "greater than or equal to" or "less than or equal to". \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) \( \newcommand{\dsum}{\displaystyle\sum\limits} \) \( \newcommand{\dint}{\displaystyle\int\limits} \)

\( \newcommand{\dlim}{\displaystyle\lim\limits} \) It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate, items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts. But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition.

However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations. Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further. Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers.

In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions. The properties of the Real Number System will prove useful when working with equations, functions and formulas in Algebra, as they allow for the creation of equivalent expressions which will often aid in solving... In addition, they can be used to help explain or justify solutions. Don't panic!!! You already know the "mathematics" at work in this chart. The chart just tells you the "fancy names" for these different math facts.

Most of these properties are just common sense. The first 13 properties are the most common, and need to be known by "name". The remaining properties, referred to as the "equality properties of numbers", will be used to explain steps when solving equations. FYI: Regarding the term "natural numbers", there is no universal agreement about whether to include zero in this set. Most mathematicians hold with the older traditional and define natural numbers to be the counting numbers [positive integers {1, 2, 3, ...}]. Computer scientists, set theorists, logicians and other mathematicians define natural numbers to be the whole numbers [non-negative integers {0, 1, 2, ...}].

This site will use the term "natural numbers" to refer to the counting numbers {1, 2, 3, ...}. Topical Outline | Algebra 1 Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts

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It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed len...

They First Used Them To Show Reciprocals. Later, They Used

They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts. But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null co...

Clearly, There Was Also A Need For Numbers To Represent

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further. Because of the evolution of the number system, we can now perform complex calculations using these and other categories of rea...

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The Set Of Integers Adds The Opposites Of The Natural

The set of integers adds the opposites of the natural numbers to the set of whole numbers: {…,-3, -2, -1, 0, 1, 2, 3,…}. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the in...