The Rules of Signed Numbers

Oct 13, 2025 by miodeya.com 64 views

Introduction

In mathematics, signed numbers are a fundamental concept that plays a crucial role in various mathematical operations, including multiplication. A signed number is a number that has a positive or negative value. When we multiply two signed numbers, the result can be either positive or negative, depending on the signs of the numbers being multiplied. In this article, we will explore how to determine whether the product of two signed numbers is positive or negative.

The Rules of Signed Numbers

Before we dive into the examples, let's review the basic rules of signed numbers:

A positive number multiplied by a positive number is positive.
A positive number multiplied by a negative number is negative.
A negative number multiplied by a negative number is positive.
A negative number multiplied by a positive number is negative.

Example 1: Multiplying a Negative Fraction by a Positive Fraction

Let's consider the first example:

$\left(-\frac{2}{3}\right)\left(\frac{52}{4}\right)$

To determine whether the product is positive or negative, we need to follow the rules of signed numbers. Since the first number is negative and the second number is positive, the product will be negative.

Step 1: Multiply the Numerators

The numerators are 2 and 52. Multiplying them together, we get:

$2 \times 52 = 104$

Step 2: Multiply the Denominators

The denominators are 3 and 4. Multiplying them together, we get:

$3 \times 4 = 12$

Step 3: Write the Product

Now that we have the product of the numerators and the product of the denominators, we can write the product as:

$\frac{104}{12}$

Step 4: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of 104 and 12. The GCD is 4. Dividing both the numerator and the denominator by 4, we get:

$\frac{104}{12} = \frac{26}{3}$

Conclusion

The product of $\left(-\frac{2}{3}\right)\left(\frac{52}{4}\right)$ is $\frac{26}{3}$ , which is a positive number.

Example 2: Multiplying Two Positive Fractions

Let's consider the second example:

$\left(\frac{45}{9}\right)\left(\frac{4}{5}\right)$

To determine whether the product is positive or negative, we need to follow the rules of signed numbers. Since both numbers are positive, the product will be positive.

Step 1: Multiply the Numerators

The numerators are 45 and 4. Multiplying them together, we get:

$45 \times 4 = 180$

Step 2: Multiply the Denominators

The denominators are 9 and 5. Multiplying them together, we get:

$9 \times 5 = 45$

Step 3: Write the Product

Now that we have the product of the numerators and the product of the denominators, we can write the product as:

$\frac{180}{45}$

Step 4: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of 180 and 45. The GCD is 45. Dividing both the numerator and the denominator by 45, we get:

$\frac{180}{45} = \frac{4}{1}$

Conclusion

The product of $\left(\frac{45}{9}\right)\left(\frac{4}{5}\right)$ is 4, which is a positive number.

Example 3: Multiplying a Negative Fraction by a Negative Fraction

Let's consider the third example:

$\left(-\frac{3}{4}\right)\left(-\frac{6}{8}\right)$

To determine whether the product is positive or negative, we need to follow the rules of signed numbers. Since both numbers are negative, the product will be positive.

Step 1: Multiply the Numerators

The numerators are 3 and 6. Multiplying them together, we get:

$3 \times 6 = 18$

Step 2: Multiply the Denominators

The denominators are 4 and 8. Multiplying them together, we get:

$4 \times 8 = 32$

Step 3: Write the Product

Now that we have the product of the numerators and the product of the denominators, we can write the product as:

$\frac{18}{32}$

Step 4: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of 18 and 32. The GCD is 2. Dividing both the numerator and the denominator by 2, we get:

$\frac{18}{32} = \frac{9}{16}$

Conclusion

The product of $\left(-\frac{3}{4}\right)\left(-\frac{6}{8}\right)$ is $\frac{9}{16}$ , which is a positive number.

Conclusion

In this article, we have explored how to determine whether the product of two signed numbers is positive or negative. We have used three examples to illustrate the rules of signed numbers and how to multiply fractions. By following these rules and simplifying the fractions, we can determine whether the product is positive or negative.