Problem 17 - Algebra 1

Question

Prove the theorem. (Use the basic axioms of algebra and the definition of subtraction given in Example \(1 .\) ) If \(a\) is any real number, then \(-1(a)=-a\)

Step-by-Step Solution

Answer

The theorem \(-1(a) = -a\) holds true according to the basic axioms of algebra and the definition of subtraction. This is because the sum of a number and its additive inverse (negative) always equals zero. Thus, \(-1\) times any real number \(a\) equals the negative of \(a\).

Step 1: Understand the definition of subtraction

In the context of algebra, the difference of \(a\) and \(b\), \(a-b\), can be rewritten as \(a + (-b)\), where \(-b\) is the additive inverse of \(b\). This means that when you add \(b\) and \(-b\) together, you get zero.

Step 2: Apply the algebraic axiom

From the axiom, we know that \(0\) is an unique additive identity, for any real number \(a\), \(a + 0 = a\). To prove \(-1(a)=-a\), let's assume \(a\) is multiplied by \(-1\), which gives us \(-a\). Now, if we add \(a\) and \(-a\) together according to the definition of subtraction, we will get zero.

Step 3: Conclude the proof

Since the axioms guarantee that the sum of a number and its additive inverse (negative) always equals zero and we showed that the sum of \(a\) and \(-a\) equals zero, we can conclude that \(-1(a) = -a\) is a true statement.

Key Concepts

Real Numbers

Real numbers are the building blocks of many mathematical concepts. They include all numbers that can be found on the number line. This encompasses natural numbers, whole numbers, integers, rational numbers (like fractions and decimals), and irrational numbers (like square roots of non-perfect squares and pi).
Real numbers can be positive, negative, or zero. They are useful in measuring, counting, and performing mathematical calculations in everyday life.
In algebra, real numbers are often represented by variables such as \(a\) and \(b\), allowing us to solve equations and prove theorems.

Axioms of Algebra

Axioms of algebra are fundamental principles used to build and understand all of algebra. These axioms are like the rules of a game, setting the framework within which mathematical logic operates.
Some key axioms include:

• Additive Identity: For any real number \(a\), \(a + 0 = a\).
• Additive Inverse: For every real number \(a\), there exists a number \(-a\), such that \(a + (-a) = 0\).
• Distributive Property: For real numbers \(a, b, c\), \(a(b + c) = ab + ac\).

These axioms ensure consistency and reliability in proving mathematical statements and solving equations.

Additive Inverse

The concept of an additive inverse is crucial in algebra, as it allows subtraction to be viewed in terms of addition. The additive inverse of a real number \(b\) is \(-b\). Adding a number and its additive inverse results in zero, which is the basis for subtraction.
For example, if you have \(a = 5\), then its additive inverse is \(-a = -5\). When added together, \(5 + (-5) = 0\).
This property is particularly important in proofs and simplifications, making it easier to manipulate and solve algebraic expressions.

Subtraction Definition

In algebra, subtraction is defined using the concept of addition. Instead of viewing subtraction as taking away, we view it as the addition of an additive inverse.
For any real numbers \(a\) and \(b\), the difference \(a - b\) is defined as \(a + (-b)\).
The beauty of this definition is that it aligns subtraction with the known properties of addition, allowing for consistent manipulation of equations. This view supports the axioms and principles of algebra, making complex proofs, like proving \(-1(a) = -a\), understandable and logical.