Integers and Absolute Value Study Notes

Integers are whole numbers and their opposites. They include positive numbers, negative numbers, and zero. Unlike fractions or decimals, integers do not have fractional parts.

🔢 What Are Integers?

An integer is any number from the set: \( \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \). They can be:

• Positive Integers: Numbers greater than zero (e.g., \(1, 2, 3, \ldots \)). They are sometimes written with a plus sign (e.g., \(+5\)), but usually, no sign means positive.
• Negative Integers: Numbers less than zero (e.g., \( \ldots, -3, -2, -1 \)). They are always written with a minus sign (e.g., \(-7\)).
• Zero: Neither positive nor negative. It is the integer that separates positive and negative numbers.

🌍 Real-World Examples of Integers

Integers help us describe situations where quantities can be above or below a certain point:

• Temperature: \(5^\circ\)C above zero is \(+5\), \(10^\circ\)C below zero is \(-10\).
• Elevation: \(100\) feet above sea level is \(+100\), \(50\) feet below sea level is \(-50\).
• Money: Earning \( \20 \) is \(+20\), owing \( \15 \) is \(-15\).

📏 Integers on a Number Line

A number line is a visual tool to understand integers. Key features:

• Zero (0): Always in the middle.
• Positive Numbers: To the right of zero, increasing as you move right.
• Negative Numbers: To the left of zero, decreasing as you move left.

Example: Imagine a number line. \[ \ldots \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad \ldots \]

↔️ Opposites of Integers

The opposite of an integer is the number that is the same distance from zero on the number line but in the opposite direction.

• The opposite of \(5\) is \(-5\).
• The opposite of \(-8\) is \(8\).
• The opposite of \(0\) is \(0\).

Example: \(5\) is \(5\) units to the right of \(0\). Its opposite, \(-5\), is \(5\) units to the left of \(0\).

🔍 Comparing and Ordering Integers

When comparing integers, remember their position on the number line:

• The further an integer is to the right on the number line, the greater its value.
• The further an integer is to the left on the number line, the smaller its value.

Signs for Comparing

• \( > \) means "greater than"
• \( < \) means "less than"
• \( = \) means "equal to"

Examples:

• \( 3 > -5 \) (Because \(3\) is to the right of \(-5\))
• \( -2 < 1 \) (Because \(-2\) is to the left of \(1\))
• \( -4 < -1 \) (Because \(-4\) is to the left of \(-1\))
• \( 0 > -7 \) (Because \(0\) is to the right of \(-7\))

Ordering Integers

To order integers, place them on an imaginary number line and then list them from least to greatest (left to right) or greatest to least (right to left).

Example: Order the following integers from least to greatest: \( -3, 2, 0, -5, 1 \) Solution: On a number line, you would find them in this order: \(-5, -3, 0, 1, 2\). So, the ordered list is \( -5, -3, 0, 1, 2 \).

✨ Absolute Value

The absolute value of an integer is its distance from zero on the number line. Since distance is always positive (or zero), the absolute value of any non-zero integer is always positive.

Symbol for Absolute Value

We use two vertical bars to denote absolute value. For example, the absolute value of \(a\) is written as \( |a| \).

Key Points:

• The absolute value of a positive number is the number itself. Example: \( |7| = 7 \)
• The absolute value of a negative number is its positive opposite. Example: \( |-7| = 7 \)
• The absolute value of zero is zero. Example: \( |0| = 0 \)

Calculating Absolute Value

To find the absolute value, simply remove any negative sign. If there's no sign (or it's positive), the number stays the same.

Integer	Distance from Zero	Absolute Value
\( 5 \)	\( 5 \) units right	\( |5| = 5 \)
\( -5 \)	\( 5 \) units left	\( |-5| = 5 \)
\( 12 \)	\( 12 \) units right	\( |12| = 12 \)
\( -9 \)	\( 9 \) units left	\( |-9| = 9 \)
\( 0 \)	\( 0 \) units	\( |0| = 0 \)

💡 Pro Tip: Absolute Value vs. Opposite

The opposite of \(5\) is \(-5\). The absolute value of \(5\) is \(5\).
The opposite of \(-5\) is \(5\). The absolute value of \(-5\) is \(5\).
Absolute value always results in a non-negative number, while an opposite can be negative or positive.

Comparing Absolute Values

You can also compare the absolute values of integers.

• Compare \( |-3| \) and \( |2| \).
  \( |-3| = 3 \) and \( |2| = 2 \). So, \( |-3| > |2| \) because \( 3 > 2 \).
• Compare \( |-5| \) and \( |-8| \).
  \( |-5| = 5 \) and \( |-8| = 8 \). So, \( |-5| < |-8| \) because \( 5 < 8 \).