Problem 72 - Algebra 1

Question

Which of the following statements is not true? (A) The product of any number and zero is zero. (B) The order in which two numbers are multiplied does not change the product. (c) The product of any number and \(-1\) is a negative number. (D) The product of any number and \(-1\) is the opposite of the number.

Step-by-Step Solution

Answer

The false statement among the provided options is (C) 'The product of any number and -1 is a negative number'. This is not true in case of negative numbers.

Step 1: Analyse statement (A)

The statement (A) 'The product of any number and zero is zero'. This is a true statement. It's a basic multiplication property, as any number multiplied by zero equals zero.

Step 2: Analyse statement (B)

The statement (B) 'The order in which two numbers are multiplied does not change the product'. This is known as the commutative property of multiplication and is also true, as changing the order of multiplication does not affect the product. For instance, \(5 \times 3 = 15\) and \(3 \times 5 = 15\).

Step 3: Analyse statement (C)

The statement (C) 'The product of any number and -1 is a negative number'. On the surface, this statement appears to be true. However, it's crucial to consider all possible numbers, including negative ones. As, when a negative number is multiplied by -1, the result is a positive number, not a negative number. Therefore, this statement is false.

Step 4: Analyse statement (D)

The statement (D) 'The product of any number and -1 is the opposite of the number'. This is true, as multiplying any number by -1 simply changes its sign.

Key Concepts

Commutative Property

The commutative property of multiplication tells us something intriguing: it doesn't matter which way around we multiply two numbers, the result or product will always be the same. For example, if you multiply 4 and 7, it doesn't matter if you do it as \(4 \times 7\) or \(7 \times 4\), the answer is 28 either way.
This property makes calculations flexible and helps simplify solving mathematical problems, especially when dealing with larger numbers or complex equations. Here’s an easy way to remember this: think of it as numbers being friendly—they don’t mind switching places, and yet they still come up with the same answer!
This concept is at the very core of many arithmetic operations and is essential for understanding more advanced mathematics and algebra later on.

Multiplicative Identity

The concept of multiplicative identity is quite straightforward. It means that when you multiply any number by one, the number remains the same. In mathematical terms, this property is expressed as \(a \times 1 = a\).
This idea is easy to grasp. It highlights the special role of the number 1 in multiplication. It's like the "mirror" number of multiplication; no matter what number you "shine" it on, that number reflects back unchanged.
The multiplicative identity is an important tool in math because it helps maintain the value of a number when you're performing various operations. It's often used as a step in solving equations and is a fundamental part of simplifying mathematical expressions.

Multiplication with Negative Numbers

Multiplying with negative numbers is another fundamental aspect of arithmetic that often confuses. Let’s make this simple: if you multiply two negative numbers, the result is a positive number. But if you multiply a negative number by a positive one, the result will be negative.
For example, multiplying \(-2 \times -3\) gives a positive 6. On the other hand, \(-2 \times 3\) results in \(-6\). This might seem like magic at first, but it's simply about understanding patterns.
Consider multiplying a number by \(-1\). If we take any number, say 9, and multiply it by \(-1\), we get \(-9\). Basically, multiplying by \(-1\) flips the sign, making positive numbers negative, and negative ones positive.

• If the number is positive, it becomes negative.
• If the number is negative, it becomes positive.

This underlines why statement (C) from the original exercise isn't entirely true—it fails to account for multiplying negative numbers by \(-1\).