Integers: The World of Whole Numbers
What Are Integers?
Integers are a set of numbers that include all the whole numbers, both positive and negative, and zero. They do not include fractions or decimals. We use the symbol $ \mathbb{Z} $ to represent the set of all integers[1].
We can visualize integers on a number line, which extends infinitely in both the positive (right) and negative (left) directions, with zero at the center.
| Type of Integer | Description | Examples |
|---|---|---|
| Positive Integers | Numbers greater than zero. They are to the right of zero on the number line. | 1, 2, 3, 4, 5, ... |
| Negative Integers | Numbers less than zero. They are written with a minus sign (-) and are to the left of zero on the number line. | -1, -2, -3, -4, -5, ... |
| Zero | Zero is neither positive nor negative. It is the neutral integer that separates positive and negative numbers. | 0 |
The Number Line and Comparing Integers
The number line is a powerful tool for understanding integers. On a standard horizontal number line, numbers increase as you move to the right and decrease as you move to the left. This helps us compare integers easily.
Rule for Comparison: Any number to the right on the number line is greater than a number to its left. For example, $-2$ is greater than $-5$ because -2 is to the right of -5. We write this as $-2 > -5$.
The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always positive or zero. The absolute value of a number $a$ is written as $|a|$. For example, $|5| = 5$ and $|-5| = 5$.
Adding and Subtracting Integers
Performing operations with integers can be tricky at first, but a few simple rules make it easy.
Addition
When adding integers, the signs of the numbers determine the direction you move on the number line.
- Same Signs: Add the absolute values and keep the common sign.
Example: $(-6) + (-3) = -9$ and $5 + 4 = 9$. - Different Signs: Subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value.
Example: $(-7) + 2 = -5$ and $8 + (-5) = 3$.
Subtraction
Subtracting an integer is the same as adding its opposite. This is a key rule to remember.
For any integers $a$ and $b$:
$a - b = a + (-b)$
Example: To calculate $5 - 8$, we change it to addition of the opposite: $5 + (-8) = -3$.
Example: To calculate $-4 - (-6)$, we change it to $-4 + 6 = 2$.
Multiplying and Dividing Integers
The rules for multiplication and division are straightforward and depend on the signs of the numbers involved.
| Operation | Rule | Example |
|---|---|---|
| Multiplication $(\times)$ | Positive $\times$ Positive = Positive Positive $\times$ Negative = Negative Negative $\times$ Positive = Negative Negative $\times$ Negative = Positive | $4 \times 5 = 20$ $3 \times (-4) = -12$ $-2 \times 6 = -12$ $-3 \times (-5) = 15$ |
| Division $(\div)$ | Positive $\div$ Positive = Positive Positive $\div$ Negative = Negative Negative $\div$ Positive = Negative Negative $\div$ Negative = Positive | $15 \div 5 = 3$ $20 \div (-4) = -5$ $-18 \div 3 = -6$ $-24 \div (-8) = 3$ |
In short, if the signs are the same, the answer is positive. If the signs are different, the answer is negative.
Integers in Everyday Life
Integers are not just abstract mathematical concepts; they are used all around us. Here are some common examples:
- Temperature: Weather forecasts use positive and negative integers. A temperature of $25^\circ\text{C}$ is a positive integer, while $-10^\circ\text{C}$ is a negative integer, indicating freezing conditions.
- Elevation and Depth: Sea level is often considered zero. A mountain peak might have an elevation of $+2,000$ meters, while the bottom of a valley could be at $-50$ meters. A submarine might dive to a depth of $-500$ meters.
- Finance: In banking, positive integers represent deposits or money you have (a credit of $+\$100$), while negative integers represent debts, withdrawals, or money you owe (a debit of $-\$50$).
- Sports: In golf, scores are often represented with integers. Par is zero. A birdie ($-1$) is one stroke under par, and a bogey ($+1$) is one stroke over par.
Common Mistakes and Important Questions
Q: Is zero a positive or a negative integer?
A: Zero is neither positive nor negative. It is a neutral integer that acts as the boundary between the positive and negative numbers on the number line.
Q: Why does multiplying two negative numbers give a positive answer?
A: Think of it as reversing a direction twice. If "negative" means "opposite direction," then turning around twice brings you back to the original, positive direction. For example, if you face north (positive) and a command to turn south (negative) is itself reversed (another negative), you end up facing north again (positive). Mathematically, it's a consistent rule that makes algebra work correctly.
Q: What is the difference between a negative integer and a number with a subtraction sign in front?
A: A negative integer is a number itself, like $-5$. A subtraction sign is an operation between two numbers, like in $10 - 5$. However, as we learned with the subtraction rule, $10 - 5$ is the same as $10 + (-5)$, so we can think of subtraction as adding a negative number.
Integers form a fundamental part of our numerical system, providing a way to represent quantities that can be above, below, or at a reference point like zero. Mastering the concepts of the number line, absolute value, and the rules for arithmetic operations with integers unlocks the door to more advanced mathematical topics like algebra. From checking the temperature to managing a budget, integers are deeply woven into the fabric of daily life, proving that this mathematical concept is both practical and essential.
Footnote
[1] Z (Symbol for Integers): The symbol $ \mathbb{Z} $ comes from the German word "Zahlen," which means "numbers." It represents the set of all integers: $ \{..., -3, -2, -1, 0, 1, 2, 3, ...\} $.