Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that division by 0 is meaningless as follows: Suppose that \(a \neq 0\). If \(a / 0=b\), then \(a=0 \cdot b=0\), which is a contradiction. Now find a reason why \(0 / 0\) is also meaningless.

Step-by-Step Solution

Answer

Division by zero, including \(0 / 0\), is undefined due to contradictions or ambiguity.

Step 1: Analyze Division by Zero

We begin with the division expression \(a / 0 = b\) where \(a eq 0\). By definition, division means finding a number \(b\) such that \(a = 0 \cdot b\). However, since multiplying any number by zero results in zero, we have \(a = 0\), which contradicts the initial condition \(a eq 0\). Thus, division by zero is undefined.

Step 2: Explore \(0 / 0\) Division

Now consider the expression \(0 / 0 = b\). Here we need a number \(b\) such that \(0 = 0 \cdot b\). This equation is always true for any \(b\) because multiplying zero by any number results in zero. Since \(b\) can be any value, there is no unique solution, which leads to an inherent ambiguity. Therefore, \(0 / 0\) is undefined because it does not provide a unique result.

Key Concepts

Undefined Operations

Often in mathematics, certain expressions or operations are deemed "undefined." A classic example is division by zero. When we talk about division, we usually assume that we can find a quotient or result. However, separating by a number involves dividing something into parts, and zero as a divisor doesn't make sense. Imagine trying to divide an object into zero parts; how would you even start? There's nothing to base the division on, causing the idea to be fundamentally flawed.
In the equation \(a/0 = b\), we seek a number \(b\) such that \(a = 0 \cdot b\). Multiplying any number by zero gives us zero, hence \(a\) must equal zero. This creates problems, especially when \(aeq 0\). Because this operation defies the basic principles of multiplication and division, it results in what is known as an undefined operation in mathematics.

Mathematical Contradictions

Mathematical contradictions occur when two or more statements or equations provide opposing truths or results. Recognizing these contradictions helps us detect when an operation, like division by zero, goes against established mathematical rules.
For instance, with \(a/0 = b\), we claim \(a = 0 \cdot b\). Yet, if \(a\) is non-zero, this claim cannot hold because such a multiplication would inherently result in zero. It's a logical impasse; we start by assuming \(aeq 0\) but end up proving that \(a=0\). This contradiction highlights why, under standard arithmetic rules, division by zero is not permissible.

Division Rules

To fully grasp why division by zero is undefined, it's crucial to understand the rules governing division. In standard arithmetic, division is essentially the reverse process of multiplication. When we divide \(a\) by \(b\), we ask "which number multiplied by \(b\) equals \(a\)?"
Consider what happens with division involving zero:

• If \(b\) in \(a/b\) is 0, we're stuck because any number times zero results in zero, not \(a\) unless \(a=0\).
• When \(a=0\) in the expression \(0/0 = b\), every \(b\) satisfies the multiplication \(0 = 0\times b\), producing countless potential answers. This ambiguity and lack of uniqueness make \(0/0\) undefined.

These rules reinforce the need for a non-zero divisor to obtain a single, clear result, illustrating why the practice of division by zero violates the fundamental guidelines of arithmetic.