Numbers with the Same or Opposite Signs
Two numbers are said to have the same sign if they’re both positive (e.g., 3 and 4) or both negative (e.g., -3 and -4). They have opposite signs if one is positive and the other is negative (e.g., 3 and -4).
Knowing this difference is essential, as signs play a key role in determining the results of operations, especially in multiplication and division.
Numbers with the Same Sign
Two numbers have the same sign if they’re either both positive or both negative.
For instance, 5 and 7 are both positive, so they have the same sign.
Likewise, -3 and -9 are both negative, so they also have the same sign. What matters is that the signs match.
Sign Rule for Numbers with the Same Sign
When multiplying or dividing two numbers with the same sign, the result will always be positive. For example:
\(5 \times 7 = 35\) (positive);
\((-3) \times (-9) = 27\) (positive).
This rule is fundamental in many calculations, allowing us to predict the sign of the result without doing the full operation.
Numbers with Opposite Signs
Two numbers have opposite signs when one is positive and the other is negative.
For example, 4 and -2 have opposite signs.
Another example is -8 and 5.
Sign Rule for Numbers with Opposite Signs
In multiplication and division, the result of two numbers with opposite signs will always be negative. For example:
\(4 \times (-2) = -8\) (negative);
\((-8) \div 5 = -1.6\) (negative).
This rule lets us determine the sign of the result just by looking at the signs of the numbers involved.
Why Is It Useful to Know About Same and Opposite Signs?
Recognizing whether numbers have the same or opposite signs simplifies many calculations.
In algebra, for instance, simplifying expressions often involves applying the sign rule.
| x | + | - |
|---|---|---|
| + | + | - |
| - | - | + |
This concept is also important in physics.
For example, when dealing with forces: two forces in the same direction (same sign) add together, while two forces in opposite directions (opposite signs) offset each other, potentially leading to balance or cancellation.
In summary, a number’s sign isn’t just a minor detail; it has a major impact on the results of operations and influences real-world scenarios.
