Problem 118 - Intermediate Algebra

Question

In each case, explain what you can conclude about one or both of the numbers. a. When the two numbers are added, the result is \(0 .\) b. When the two numbers are subtracted, the result is \(0 .\) c. When the two numbers are multiplied, the result is 0 . d. When the two numbers are divided, the result is 0 .

Step-by-Step Solution

Answer

a) Numbers are additive inverses. b) Numbers are equal. c) At least one number is zero. d) Numerator is zero, denominator is non-zero.

Step 1: Analyze addition result

Given that the sum of two numbers is 0, denoted as \( a + b = 0 \), this implies that the two numbers must be additives inverses of each other. Thus, one number is the negation of the other, such as \( b = -a \). If \( a = 3 \), then \( b = -3 \), ensuring the sum is zero.

Step 2: Analyze subtraction result

Given that the difference between two numbers is 0, denoted as \( a - b = 0 \), this implies that the two numbers must be equal since subtracting two equal numbers results in zero. Therefore, we conclude \( a = b \). For example, if \( a = 5 \), then \( b = 5 \).

Step 3: Analyze multiplication result

Given that the product of two numbers is 0, denoted as \( a \times b = 0 \), it implies that at least one of the numbers must be zero. This is because the only way for a product to equal zero is if one or both multiplied terms are zero. Therefore, either \( a = 0 \), \( b = 0 \), or both.

Step 4: Analyze division result

Given that the quotient of two numbers is 0, denoted as \( \frac{a}{b} = 0 \), it implies that the numerator must be zero while the denominator must be a non-zero number. Thus, \( a = 0 \) and \( b eq 0 \), because dividing zero by any non-zero number results in zero.

Key Concepts

Additive Inverses

When two numbers are added together and their sum equals zero, it means they are additive inverses of one another. The basic property of additive inverses is that the second number precisely "negates" the first. For example, if you have a number like 7, its additive inverse would be -7. Adding these together simplifies to:

• 7 + (-7) = 0

This principle is very important because it helps in simplifying expressions and solving equations. For a pair of numbers like \(a\) and \(b\), when their addtion equates to zero via \(a + b = 0\), it directly implies \(b = -a\). Each pair might look different on the surface, but this inverse property is an underlining truth. Take note that for every real number, there is always a unique additive inverse. This might be a small addition, but it's a basic yet pivotal algebraic tool which helps us solve for unknowns and verify our solutions.
Understanding additive inverses can help in breaking down complex equations into simpler parts. This way, when presented with algebraic challenges, you can easily determine missing values by leveraging this property!

Multiplicative Properties

Multiplicative properties are other fundamental principles in algebra. When we say the result of multiplying two numbers is zero, \(a \times b = 0\), there’s a significant insight we gain - at least one of those numbers must be zero. This property stems from the rule of zero product, which states when a product is zero, one or more of the multiplicands must be zero as well. Consider a simple example:

• 0 \times 4 = 0
• 7 \times 0 = 0
• 0 \times 0 = 0

Each case definitively shows that zero itself is the primary factor causing the product to zero out.
The zero product property is immensely useful while solving equations because it can instantly help determine the values of unknown variables. For instance, if you had \(x \times y = 0\), you immediately understand that either \(x\) or \(y\) (or both) must be zero to satisfy this property.
This rule profoundly simplifies problem-solving in algebra and helps students intuitively understand why multiplicative zero always points to the identity element in the scenario.

Division by Zero

One of the trickiest aspects students encounter in math is the concept of division by zero. If asked, "What happens when a number is divided and the result is zero?", Understanding Division Outcomes:When \( \frac{a}{b} = 0 \), it means you are looking at a situation where the dividend \(a\) itself is zero. The divisor \(b\), however, must not be zero because dividing by zero is undefined in mathematics. To illustrate:

• \( \frac{0}{5} = 0\)
• \( \frac{0}{-3} = 0\)
• \( \frac{0}{1} = 0\)

The common theme here is the dividend is zero, while the divisor can be any non-zero number. Remember, dividing by zero doesn't produce a meaningful value or solution in mathematical terms due to undefined rules.
This principle often confuses students, but it's fundamental to ensure calculations are accurate and logical. By really understanding these properties, students manage to avoid common pitfalls in math and increase their ease of operations with fractions and ratios.