Relative Numbers
Problem — How to understand and work with positive and negative numbers to calculate and solve problems?
- Understand what relative numbers are and their usefulness.
- Know how to compare and place relative numbers on a number line.
- Learn addition, subtraction, multiplication, and division with relative numbers.
- Solve real-life problems using relative numbers.
- Master the vocabulary and rules associated with relative numbers.
Part 1: Introduction to Relative Numbers
Relative numbers are numbers that can be positive, negative, or zero. They represent situations involving gain and loss, rise and fall, temperatures above and below zero, etc.
For example, if the temperature is -3 °C, it is below zero. If a player lost 5 points, it is written as -5. Relative numbers are written with a + sign for positives (often optional) and a - sign for negatives.
The Number Line and Relative Numbers
We can represent relative numbers on a number line where zero is the central point. Positive numbers are placed to the right of zero, negative numbers to the left.
- The further right you go, the larger the numbers become.
- The further left you go, the smaller the numbers become (more negative).
Relative numbers extend natural numbers by allowing expression of negative quantities or values below a reference point like zero. The number line is an essential tool to visualize and compare these numbers. Understanding their notation and placement is the foundation for learning to work with them.
Part 2: Comparing and Ordering Relative Numbers
Comparing two relative numbers means determining which one is greater or smaller. This comparison is done using their positions on the number line.
A number is greater than another if it is located to the right on the number line. It is smaller if it is to the left.
Example: Let's compare -4 and 2. On the number line, -4 is left of 0 and 2 is right of 0, so 2 > -4.
Tips for Comparing
- Any positive number is always greater than any negative number.
- Between two positive numbers, the one with the larger absolute value is greater (e.g., 5 > 3).
- Between two negative numbers, the one with the smaller absolute value is greater (e.g., -3 > -5).
Reminder: The absolute value of a relative number is its distance from zero. For example, the absolute value of -7 is 7, and the absolute value of +7 is also 7.
Comparing relative numbers requires understanding their position on the number line well. It is essential to remember that positives are always greater than negatives and that for negatives, the usual order is reversed: the further left you go from zero, the smaller the number.
Part 3: Addition and Subtraction of Relative Numbers
Adding or subtracting relative numbers follows rules that take the signs of the numbers into account.
Addition:
- If both numbers have the same sign, add their absolute values and keep the sign.
- If the numbers have different signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value.
Examples:
- +5 + +3 = +(5 + 3) = +8
- -7 + -2 = -(7 + 2) = -9
- +6 + (-4) = +(6 - 4) = +2
- -5 + (+8) = +(8 - 5) = +3
Subtraction: Subtracting a number is the same as adding its opposite.
For example, 7 - (+3) = 7 + (-3) = 4 and 5 - (-2) = 5 + (+2) = 7.
Addition and subtraction operations are simplified by the concept of opposites: subtracting a number is equivalent to adding its inverse. Understanding the rules requires analyzing signs and absolute values, enabling accurate calculation with relative numbers.
Part 4: Multiplication and Division of Relative Numbers
The rules for multiplying or dividing relative numbers depend on the signs of the numbers involved.
- Multiplying or dividing two numbers with the same sign gives a positive result.
- Multiplying or dividing two numbers with different signs gives a negative result.
Examples:
- (+4) × (+3) = +12
- (-5) × (-2) = +10
- (+6) ÷ (-3) = -2
- (-8) ÷ (+4) = -2
First multiply or divide the absolute values, then apply the sign according to the rules above.
Multiplication and division with relative numbers rely on the concept of signs. Knowing these rules is essential for solving calculations by first multiplying or dividing the absolute values and then applying the correct sign. This also helps to check and simplify expressions.
Part 5: Applications and Problems with Relative Numbers
Relative numbers are used in many real-life situations: temperature, altitude, gains and losses, debts, etc. Knowing how to handle them helps solve diverse problems.
Example:
A diver is 5 meters below sea level (-5 m). He descends 3 meters, then ascends 7 meters. What is his final position relative to sea level?
Solution:
- Initial position: -5 m
- He goes down 3 m ⇒ add -3 m: -5 + (-3) = -8 m
- He goes up 7 m ⇒ add +7 m: -8 + 7 = -1 m
The diver is therefore 1 meter below sea level after these movements.
Another Problem:
The temperature is -2 °C in the morning, it drops by 5 °C at night, then rises by 8 °C in the afternoon. What is the temperature at the end of the day?
Solution:
- Initial temperature: -2 °C
- Drop of 5 °C: -2 + (-5) = -7 °C
- Rise of 8 °C: -7 + 8 = +1 °C
Real-life problems help understand the usefulness of relative numbers. Translating situations into calculations with positive and negative numbers requires careful identification of changes and choosing the correct signs. This step is crucial to correctly apply addition and subtraction rules for relative numbers.
Relative numbers are an essential extension of positive numbers that allow expression of quantities less than a reference like zero. Understanding them involves mastering the number line, comparison, and especially specific rules for addition, subtraction, multiplication, and division. These rules are fundamental for solving problems in various contexts, both mathematical and everyday. A good command of relative numbers paves the way to more advanced concepts and provides a solid foundation for future mathematics learning.