Negative Integers: The World Below Zero
What Are Negative Integers?
Negative integers are whole numbers that are less than zero. They are written with a minus sign (-) in front of a positive integer. For example, -1, -5, and -100 are all negative integers. Together with positive integers and zero, they form the set of integers, which is often represented as {..., -3, -2, -1, 0, 1, 2, 3, ...}.
The concept of numbers less than zero was not always accepted. Ancient civilizations like the Egyptians and Babylonians had no use for them. The first recorded use of negative numbers was in ancient China around 200 BCE in the Nine Chapters on the Mathematical Art, where they used black rods for positive numbers and red rods for negative numbers. In India, around 600-700 CE, mathematicians like Brahmagupta established rules for arithmetic with negative numbers. European mathematicians were much slower to adopt them, often referring to them as "absurd" or "fictitious" numbers until the 17th century.
The number line is a powerful visual tool for understanding integers. Zero is at the center. Positive integers extend to the right, and negative integers extend to the left. The number line shows that for every positive integer, there is a corresponding negative integer, known as its opposite. For example, the opposite of 3 is -3, and the opposite of -7 is 7.
Absolute Value and Comparing Integers
The absolute value of a number is its distance from zero on the number line, regardless of direction. Distance is always a positive number (or zero). The absolute value of a number n is written as $|n|$.
For example:
- $|5| = 5$ because 5 is 5 units from zero.
- $|-5| = 5$ because -5 is also 5 units from zero.
- $|0| = 0$.
When comparing two integers on a number line, the number to the right is always the greater number. This leads to the following rules:
- Any positive integer is greater than any negative integer.
- Zero is greater than any negative integer.
- When comparing two negative integers, the one closer to zero is greater. For example, $-10 < -1$ because -10 is further to the left on the number line.
Arithmetic with Negative Integers
Performing calculations with negative integers follows specific, consistent rules. Let's break them down step by step.
Addition
Adding integers can be visualized as moving along the number line.
- Adding a positive integer: Move to the right.
$ -3 + 5 = 2 $ (Start at -3, move 5 steps right to land on 2). - Adding a negative integer: Move to the left.
$ 4 + (-6) = -2 $ (Start at 4, move 6 steps left to land on -2). - Adding two negative integers: Move to the left, and the sum is negative.
$ -2 + (-4) = -6 $ (Start at -2, move 4 steps left to land on -6).
Subtraction
Subtraction is the same as adding the opposite. This is a fundamental rule that simplifies calculations.
To subtract an integer, add its opposite:
$ a - b = a + (-b) $
Examples:
- $ 7 - 3 = 7 + (-3) = 4 $
- $ 5 - (-2) = 5 + 2 = 7 $ (The opposite of -2 is 2).
- $ -4 - 3 = -4 + (-3) = -7 $
- $ -1 - (-5) = -1 + 5 = 4 $
Multiplication
The rules for multiplication depend on the signs of the numbers being multiplied.
| Sign of First Number | Sign of Second Number | Result | Example |
|---|---|---|---|
| Positive (+) | Positive (+) | Positive (+) | $ 3 \times 4 = 12 $ |
| Positive (+) | Negative (-) | Negative (-) | $ 5 \times (-2) = -10 $ |
| Negative (-) | Positive (+) | Negative (-) | $ (-6) \times 3 = -18 $ |
| Negative (-) | Negative (-) | Positive (+) | $ (-4) \times (-5) = 20 $ |
A simple way to remember this is: Like signs give a positive product; unlike signs give a negative product.
Division
Division follows the same sign rules as multiplication.
| Sign of First Number | Sign of Second Number | Result | Example |
|---|---|---|---|
| Positive (+) | Positive (+) | Positive (+) | $ 12 \div 4 = 3 $ |
| Positive (+) | Negative (-) | Negative (-) | $ 15 \div (-3) = -5 $ |
| Negative (-) | Positive (+) | Negative (-) | $ (-18) \div 6 = -3 $ |
| Negative (-) | Negative (-) | Positive (+) | $ (-24) \div (-8) = 3 $ |
Remember: Like signs give a positive quotient; unlike signs give a negative quotient. Also, division by zero is undefined.
A simple phrase to remember the rules for multiplication and division is: "Same signs, positive. Different signs, negative."
Negative Integers in the Real World
Negative numbers are not just abstract mathematical concepts; they are used everywhere in daily life and science.
- Temperature: This is the most common example. On a Celsius or Fahrenheit scale, temperatures below freezing are represented by negative numbers. For instance, -5°C is 5 degrees below zero.
- Finance and Debt: In banking, a negative account balance represents debt or money owed. If you have -$50 in your account, it means you owe the bank $50. Profits can be positive, while losses are negative.
- Elevation and Geography: Elevations below sea level are denoted by negative integers. For example, Death Valley in California is approximately -86 meters relative to sea level.
- Sports and Games: In golf, scores under par are negative. In some card games, points can be deducted, leading to a negative score.
- Science and Engineering: Negative numbers are used to represent forces in opposite directions, electric charge (electrons have a negative charge), and positions in a coordinate system.
Common Mistakes and Important Questions
Q: Is zero a positive or negative integer?
A: Zero is neither positive nor negative. It is a neutral integer that separates the positive and negative numbers on the number line.
Q: Why does multiplying two negative numbers give a positive number?
A: This can be understood through patterns. Look at this sequence:
$ 3 \times (-2) = -6 $
$ 2 \times (-2) = -4 $
$ 1 \times (-2) = -2 $
$ 0 \times (-2) = 0 $
Notice the product is increasing by 2 each time. To continue the pattern:
$ (-1) \times (-2) = 2 $
$ (-2) \times (-2) = 4 $
This pattern consistency forces the product of two negatives to be positive. It also makes the rules for multiplication work logically with other mathematical properties.
Q: What is the common mistake when subtracting negative integers?
A: The most common mistake is forgetting that subtracting a negative is the same as adding a positive. For example, in $ 8 - (-3) $, many students incorrectly write $ 8 - 3 = 5 $. The correct calculation is $ 8 - (-3) = 8 + 3 = 11 $. Always remember: "Two negatives make a positive" in subtraction.
Negative integers, the numbers less than zero, are a fundamental and indispensable part of mathematics. From their historical origins to their practical applications in temperature, finance, and science, they provide a complete system for representing quantities in opposite directions. Mastering the rules for their arithmetic operations—especially the key principle that subtracting a negative is equivalent to addition, and that the product or quotient of two negatives is positive—opens the door to understanding more advanced mathematical concepts. By visualizing them on the number line and applying them to real-world contexts, negative numbers become intuitive and powerful tools for problem-solving.
Footnote
1 Integer: A whole number that can be positive, negative, or zero.
2 Absolute Value: The non-negative value of a number without regard to its sign; its distance from zero.
3 Opposite: Two numbers that are the same distance from zero on the number line but on opposite sides. The sum of a number and its opposite is zero.