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In mathematics, integers are a fundamental concept that encompasses **whole numbers**, positive numbers, negative numbers, and **zero**. They play a crucial role in various mathematical operations and have applications in real-world scenarios. In this guide, we will provide a comprehensive overview of integers, including their definition, properties, and operations.

An integer is a **whole number** that includes both positive and negative numbers, as well as **zero**. Unlike decimal numbers or fractions, integers are not expressed with a fractional or decimal component. Instead, they represent whole quantities or counts.

Let’s take a look at some examples of integers: -32, -2, 0, 17, and 2,035. These numbers are considered integers because they do not contain any fractional or decimal parts. Conversely, numbers like π, 2/3, and 0.478 are not integers since they possess fractional or decimal components.

To visualize integers, imagine them as objects that cannot be divided into smaller pieces, much like a whole egg in a basket. Integers can be represented on a **number line**, a graphical representation of numbers where positive numbers are located to the right of **zero**, and negative numbers are situated to the left of zero.

Positive numbers on the **number line** increase in value as they move farther to the right, while negative numbers decrease in value as they move farther to the left. It’s important to note that a positive number is always greater in value than a negative number.

Understanding integers is essential for solving math problems, particularly in standardized tests like the ACT. Integer questions on the ACT often involve word problems and equations where variables are replaced with integers. Some questions also require an understanding of concepts such as even and odd numbers.

Integers can be subjected to various operations, including addition, subtraction, multiplication, and division. Each operation follows specific rules and guidelines. For instance, adding two integers with the same sign results in a sum with the same sign, while adding two integers with different signs involves subtracting the integers and using the sign of the larger value.

Zero is a unique integer that is neither positive nor negative. It serves as a neutral value and plays a crucial role in **integer operations** and real-world applications. For instance, integers are commonly used to represent temperatures, scores in sports games, and bank account balances.

Now that we have a basic understanding of what integers are and their significance, let’s delve deeper into their origin, types, and operations in the following sections.

## The Origin and Types of Integers

The concept of integers has a rich historical development that traces back to ancient civilizations such as the Babylonians, Mayans, and Indians. These ancient societies recognized the need for numbers that encompassed both positive and negative values, leading to the development of the integer concept.

The term “integer” itself derives from the Latin words “in” and “tangere,” which mean “untouched” or “complete.” This reflects the idea that integers are **whole numbers** that include both positive and negative values, as well as the neutral value of zero.

Furthermore, the concept of zero as a placeholder was introduced by these ancient civilizations, providing a crucial foundation for the integer system we use today. Zero serves as a reference point between positive and negative numbers, and it plays a significant role in various mathematical operations.

The development of integers as a formal mathematical concept continued over the centuries, with contributions from mathematicians such as Richard Dedekind, a German mathematician. In 1888, Dedekind created the symbol for integers, based on the German word for numbers, “Zahlen.”

The inclusion of negative numbers into the number system came later as mathematicians like Auguste-Louis Cauchy, John Herschel, and John Wallis made significant strides in the 19th century. Their work expanded the number system to incorporate negative values, giving rise to the full spectrum of integers encompassing positive numbers, negative numbers, and zero.

### Types of Integers

Integer Type | Definition |
---|---|

Positive Integers |
Whole numbers greater than zero and extending indefinitely. |

Negative Integers |
Whole numbers less than zero, denoted with a negative sign. |

Zero | A special integer that is neither positive nor negative, serving as a neutral value. |

**Positive integers** represent quantities greater than zero, such as 1, 2, 3, and so on. They form an infinite sequence that extends towards positive infinity.

**Negative integers**, on the other hand, are whole numbers less than zero. They are denoted with a negative sign, such as -1, -2, -3, and so forth. Like **positive integers**, **negative integers** form an infinite sequence towards negative infinity.

Zero holds a unique position among integers. It is neither positive nor negative, acting as a numerical placeholder and a reference point between the positive and negative number lines.

The **origin of integers** and their subsequent development throughout history is a testament to the human quest for understanding and representing mathematical concepts. The concept of integers, with their positive and negative values and the inclusion of zero, has become an essential foundation for various mathematical disciplines and real-world applications.

## Rules and Operations with Integers

When performing operations with integers, it is important to understand the rules for adding, subtracting, multiplying, and dividing. These operations allow us to work with integers and solve math problems efficiently. Let’s explore the key rules for each operation:

**Adding integers**: When adding two integers with the same sign, the sum will have the same sign as the addends. For example, -3 + (-5) = -8. However, when adding two integers with different signs, subtract the integers and use the sign of the larger value. For instance, 4 + (-7) = -3.

**Subtracting integers**: Subtraction involves changing the sign of the number being subtracted and using addition rules. This can be done vertically by aligning the numbers and performing addition with the opposite sign. For example, 5 – (-2) = 5 + 2 = 7.

Multiplying and **Dividing integers**: Multiplying two integers with the same sign results in a positive product, while multiplying two integers with different signs results in a negative product. Similarly, division follows similar rules as multiplication, with the quotient being positive when the signs are the same and negative when the signs are different.

**Absolute value**: The **absolute value** of a number is its numerical value without considering its sign. It is always positive, except for zero, and is represented by two vertical lines around the number. For example, |3| = 3 and |-4| = 4.

Mastering these rules and operations with integers is essential for solving math problems and real-world applications. Whether it’s adding, subtracting, multiplying, or dividing, understanding the rules allows us to manipulate integers with confidence and accuracy.

## FAQ

### What is an integer?

An integer is a whole number that includes positive numbers, negative numbers, and zero.

### What are examples of integers?

Examples of integers include -32, -2, 0, 17, and 2,035.

### How are integers represented on a number line?

On a **number line**, positive numbers are to the right of zero, and negative numbers are to the left of zero.

### What are the rules for adding integers?

Adding two integers with the same sign results in a sum with the same sign. Adding two integers with different signs requires subtracting the integers and using the sign of the larger value.

### How are integers subtracted?

**Subtracting integers** involves changing the sign of the number being subtracted and using addition rules.

### What are the rules for multiplying integers?

Multiplying positive integers or negative integers results in a positive product. Multiplying an odd number of negative signs produces a negative product, while an even number of negative signs produces a positive product.

### What are the rules for dividing integers?

Dividing two integers with the same sign results in a positive quotient, while dividing two integers with different signs results in a negative quotient.

### What is the absolute value of a number?

The **absolute value** of a number is its numerical value without considering its sign.

### How do operations with integers work?

Performing operations with integers involves applying the rules for adding, subtracting, multiplying, and **dividing integers**.