Problem 4
Question
Multiplying Signed Numbers.$$(-41) \times(-22)$$
Step-by-Step Solution
Verified Answer
The product of (-41) and (-22) is 902.
1Step 1: Understand the Rules for Multiplying Signed Numbers
When multiplying numbers, the product of two numbers with the same sign is positive, and the product of two numbers with different signs is negative. Since both numbers (-41) and (-22) have the same sign (negative), their product will be positive.
2Step 2: Multiply the Absolute Values
Ignore the signs of both numbers for a moment and multiply their absolute values. The absolute value of (-41) is 41 and the absolute value of (-22) is 22. Now multiply the numbers: 41 x 22.
3Step 3: Calculate the Product
Calculate 41 multiplied by 22, which equals 902.
4Step 4: Apply the Sign Rule
Since both original numbers were negative and the rule states that a negative times a negative is a positive, the final answer is positive 902.
Key Concepts
Rules of Sign for MultiplicationAbsolute ValuePositive Product Rule
Rules of Sign for Multiplication
Understanding the Rules of Sign for Multiplication is fundamental when dealing with signed numbers. The results hinge on whether the numbers involved share the same sign or have different signs. It's a concept that is quite simple, yet vital.
When multiplying two negative numbers, such as in the case with \( -41 \times -22 \), it's crucial to remember that a negative times a negative gives a positive result. Similarly, if we were multiplying two positive numbers, we would also arrive at a positive product. This is often summarized as 'like signs yield a positive product'.
On the other hand, multiplying numbers with unlike signs (one positive and one negative) results in a negative product. This could be memorized as 'unlike signs yield a negative product'. Many students find it helpful to think of this in terms of conflict or agreement: if the signs agree (they're the same), the result is positive; if they disagree (they're different), the result is negative.
When multiplying two negative numbers, such as in the case with \( -41 \times -22 \), it's crucial to remember that a negative times a negative gives a positive result. Similarly, if we were multiplying two positive numbers, we would also arrive at a positive product. This is often summarized as 'like signs yield a positive product'.
On the other hand, multiplying numbers with unlike signs (one positive and one negative) results in a negative product. This could be memorized as 'unlike signs yield a negative product'. Many students find it helpful to think of this in terms of conflict or agreement: if the signs agree (they're the same), the result is positive; if they disagree (they're different), the result is negative.
Absolute Value
The Absolute Value of a number is its distance from zero, without regard to which side of the zero it lies. It's always positive, because distances can't be negative. When multiplying, you often ignore the signs initially and focus on calculating the absolute value of each number.
For example, in our exercise \( (-41) \times (-22) \) the absolute values are 41 and 22, respectively. By focusing on these 'neutral' values, you simplify the process to a basic multiplication of two positive numbers.
In real-life terms, you might think of absolute value as the 'pure magnitude' of something. It's like measuring a distance without worrying if you're going north or south. Only after getting the multiplicative result of these absolute values (41 * 22), do you apply the sign rules to determine if your final answer is in the positive or negative territory.
For example, in our exercise \( (-41) \times (-22) \) the absolute values are 41 and 22, respectively. By focusing on these 'neutral' values, you simplify the process to a basic multiplication of two positive numbers.
In real-life terms, you might think of absolute value as the 'pure magnitude' of something. It's like measuring a distance without worrying if you're going north or south. Only after getting the multiplicative result of these absolute values (41 * 22), do you apply the sign rules to determine if your final answer is in the positive or negative territory.
Positive Product Rule
The Positive Product Rule is a specific aspect of the overall rules for multiplying signed numbers. It states that the product of two numbers with the same sign will always be positive. This applies regardless of whether the numbers are negative or positive.
In our specific exercise where we multiplied \( -41 \times -22 \), we applied this rule: since both numbers have negative signs, the result according to the Positive Product Rule would be a positive number.
This rule simplifies calculations and helps prevent errors. It's a reliable principle that can guide you through complex equations, ensuring that you always know the sign of your product when the multiplicands have the same sign. When both are negative, as many students are surprised to find, 'two wrongs do make a right' at least in the world of multiplying signed numbers!
In our specific exercise where we multiplied \( -41 \times -22 \), we applied this rule: since both numbers have negative signs, the result according to the Positive Product Rule would be a positive number.
This rule simplifies calculations and helps prevent errors. It's a reliable principle that can guide you through complex equations, ensuring that you always know the sign of your product when the multiplicands have the same sign. When both are negative, as many students are surprised to find, 'two wrongs do make a right' at least in the world of multiplying signed numbers!
Other exercises in this chapter
Problem 4
Dividing Signed Numbers Divide, keeping the proper sign on your answer. $$(-49) \div(-7)$$
View solution Problem 4
Evaluate each expression. Retain the proper number of significant digits in your answer. Powers. $$1^{3}$$
View solution Problem 4
Symbols of Equality and Inequality Insert the proper symbol of equality or inequality \((=, \approx,>,
View solution Problem 4
Adding and Subtracting Signed Numbers Combine as indicated. $$-207+(-819)$$
View solution