The Mirror World of Numbers

For thousands of years, humans only counted what they could see: one cow, two stones, three days. But what happens when you owe someone a cow? Or when the temperature drops below freezing? To describe these concepts, mathematics had to expand beyond counting numbers.

This led to the discovery of Integers. An integer is a number with no fractional part (no decimals). It includes the counting numbers (1, 2, 3), zero (0), and the negative numbers (-1, -2, -3).

1. Defining the Set

The set of integers is represented by the symbol Z. This comes from the German word Zahlen, which simply means "numbers."

Notice the ellipses (...) on both sides. This means the set goes on forever in both the positive and negative directions.

2. The Number Line

The best way to understand integers is to visualize them on a line.

[Image of integer number line]

• Zero is in the middle. It is neutral - neither positive nor negative.
• Positive Integers are to the right of zero. Their value increases as you move right.
• Negative Integers are to the left of zero. Their value decreases as you move left.

Think of it like a mirror. The number -5 is the mirror image of 5, sitting the exact same distance away from zero but in the opposite direction.

3. Absolute Value

Sometimes, we care about the "magnitude" of a number rather than its direction. This is called Absolute Value. It asks: "How far is this number from zero?"

[Image of absolute value on number line]

We write this using vertical bars: |x|.

• |5| = 5 (5 is five steps away from zero)
• |-5| = 5 (-5 is also five steps away from zero)

Absolute value is always positive (or zero), because distance cannot be negative.

4. The Rules of Engagement

When you start mixing positive and negative numbers, the rules of arithmetic change slightly. Mastering these rules is the key to algebra.

Addition

• Same Signs: Add the numbers and keep the sign.
  (+3) + (+4) = +7
  (-3) + (-4) = -7
• Different Signs: Subtract the smaller number from the larger number, and keep the sign of the larger number.
  (-10) + (+4) = -6 (because 10 is bigger than 4, and it is negative).

Subtraction

Subtraction can be tricky with negatives. The golden rule is: "Keep Change Change."

To subtract an integer, you add its opposite.

Multiplication and Division

The rules here are surprisingly simple:

• Same Signs = Positive: (+ * + = +) and (- * - = +).
• Different Signs = Negative: (+ * - = -) and (- * + = -).

Example: (-5) * (-2) = +10. (A negative times a negative is a positive).

5. History: The "False" Numbers

For a long time, mathematicians hated negative numbers. In ancient Greece, they were considered "absurd." How can you have less than nothing? Even as late as the 1700s, some European mathematicians called them "false numbers."

It was simple economics (debt) and nature (temperature) that forced the world to accept them. If you have $5 and owe someone $10, your net worth is -$5. The math had to reflect reality.

6. Real-World Applications

Integers are not just abstract concepts; they describe the world around us.

[Image of sea level elevation diagram]

1. Elevation

Geographers use Sea Level as "Zero." Mountains have positive elevation (Mount Everest is +8,848 meters). Ocean trenches have negative elevation (The Mariana Trench is -10,994 meters).

2. Temperature

In the Celsius scale, 0 degrees is the freezing point of water. If it gets colder, we enter the negative integers. A temperature of -10 degrees is ten degrees below freezing.

3. Finance and Stock Market

Profit is positive. Loss is negative. If a company makes $1 million, it is +1M. If it loses $1 million, it is -1M. Stock tickers use green (positive) and red (negative) integers to show daily changes.

7. Conclusion

Integers double the size of our mathematical universe. They allow us to move not just forward, but backward; not just up, but down. By understanding the balance between positive and negative, we gain the ability to describe opposite forces in nature, finance, and physics.