Integers and Their Properties

Integers are a set of numbers that include negative numbers, zero, and positive numbers. The set of integers is represented on the number line and allows us to compare the magnitude of numbers. Since the set of integers includes both positive and negative numbers, we consider not only their magnitude but also their sign.

The set of integers can be expressed as:
Set of Integers: { …, -3, -2, -1, 0, 1, 2, 3, … }

To understand this set better, we can examine it on the number line.

The Role of Zero

Zero is a fundamental element of integers. It is neither positive nor negative, which gives it a special place among numbers. The opposite of zero is itself. In other words, zero does not have an opposite because the concept of opposition applies only to nonzero numbers.

Zero also plays a crucial role in mathematical operations. For example:

• Multiplying any number by zero results in zero.
• Adding zero to any number does not change the value of that number: $0+x=x$.

Opposites and Comparison of Integers

Opposite Numbers

The opposite of any number is the negative of that number. For example, the opposite of 2 is -2. Mathematically, the opposite of a number expresses the same distance from zero but in the opposite direction.

Comparison of Integers

When comparing integers, placing them on the number line is very useful. The number line is a linear representation where the magnitude and relative position of numbers are clearly shown.

• Positive numbers are located to the right of zero.
• Negative numbers are located to the left of zero.
• Zero acts as a neutral point.

Example 1:

On the number line, we can see that 3 is greater than -4 because 3 is on the right and -4 is on the left.
Result: 3>−4

Example 2:

When comparing -2 and -7, we observe that -2 is to the right of -7 on the number line, meaning -2 is greater.
Result: −2>−7

Order of Operations for Calculations

To perform calculations involving integers, follow these steps:

Step 1: Solve all operations inside parentheses first. Parentheses include round brackets (), square brackets , braces { }, and absolute values | |.

Step 2: Perform exponentiation (power) operations next.

Step 3: Carry out multiplication and division from left to right in order.

Step 4: Lastly, perform addition and subtraction from left to right in order

Absolute Value

The absolute value of a number represents its distance from zero on the number line. A key feature of absolute value is that it disregards the negative sign, showing the magnitude of the number as a positive value.

Definition of Absolute Value

The absolute value of a number shows how far that number is from zero on the number line.

• For positive numbers: The absolute value of a number is the number itself. For example, the absolute value of 8 is 8 because 8 is already positive.
• For negative numbers: The absolute value of a number is the positive counterpart of that number. For example, the absolute value of -5 is 5 because -5 is 5 units away from zero.
• For zero: The absolute value of zero is zero. That is, ∣0∣=0

Rules of Absolute Value

1. Positive numbers: If the number is positive, its absolute value is the number itself.
   Example: ∣8∣=8
2. Negative numbers: If the number is negative, its absolute value is the positive counterpart of the number.
   Example: ∣−5∣=5
3. Zero: The absolute value of zero is zero.
   Example: ∣0∣=0

Uses of Absolute Value

The absolute value is often used to calculate the distance between two numbers. It is especially helpful when working with negative numbers, where the negative signs need to be removed to focus on magnitude.

Source

Sobecki, David, and Allan Bluman. Math in Our World, McGraw-Hill Education