Problem 14 - Elementary Algebra

Question

What property of real numbers justifies \(a(b+c)=(b+c) a ?\)

Step-by-Step Solution

Answer

Answer: The commutative property of multiplication justifies the formula \(a(b+c) = (b+c)a\).

Step 1: Identify the property of real numbers

The property that justifies \(a(b+c) = (b+c)a\) is the commutative property of multiplication. The commutative property states that for any real numbers a and b, the order in which we multiply them doesn't affect the result, i.e., \(ab=ba\).

Step 2: Explain the commutative property of multiplication with an example

Let's take an example with real numbers to demonstrate the commutative property of multiplication in action. Let \(a=2\), \(b=3\), and \(c=4\). According to the commutative property of multiplication, we know that \(a(b+c) = (b+c)a\). Calculating the expressions on both sides of the equation: Left Side: \(a(b+c) = 2(3+4) =2(7) = 14\). Right Side: \((b+c)a = (3+4)2 = (7)2 = 14\). As we see, the results are the same (\(14\)) on both sides, which confirms that the commutative property is indeed applied in this case.

Key Concepts

Real Numbers

When we talk about real numbers, we are referring to all the numbers that exist on the number line. This includes both rational numbers (such as fractions and integers) and irrational numbers (like \( \sqrt{2} \) or \( \pi \) ).

Real numbers are fundamental in math because they represent quantities that can be measured in the real world—distances, weights, temperatures, and more are described using real numbers. Understanding real numbers is essential because they form the basis for more complex concepts in algebra and calculus.

One of the reasons why real numbers are so powerful is that they have well-defined properties. These properties make it easier for us to work with numbers in various math operations, including multiplication, which is deeply connected to both the commutative property and algebraic expressions.

Properties of Multiplication

Multiplication has several properties that help us manipulate and understand algebraic expressions, and one of the fundamental properties is the commutative property. This property, as shown in the exercise, states that numbers can be multiplied in any order without changing the product. In simpler terms, if you have two real numbers, say \( a \) and \( b \) then \( a \times b = b \times a \).

The commutative property of multiplication helps to simplify calculations and allows us to rearrange terms for convenience without affecting the result. Besides the commutative property, there are other properties like the associative property, which deals with grouping of numbers, and the distributive property, which connects multiplication with addition and subtraction. These properties serve as tools to solve algebraic expressions efficiently and understand the structure of mathematical equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( a \) or \( b \)), and arithmetic operations (such as addition, subtraction, multiplication, and division). For example, \( 2x + 3y - 5 \) is an algebraic expression representing a quantity that can change depending on the values of \( x \) and \( y \).

Understanding how to work with algebraic expressions is crucial when solving equations and modeling real-world problems. The properties of multiplication, including the commutative property, play a significant role in rearranging and simplifying expressions. They allow mathematicians and students alike to maneuver through complex algebraic manipulations with increased flexibility and ease, leading to understanding equations and eventually solving them.

For example, given the property \( a(b + c) = (b + c)a \), students can understand that the multiplication of variables within an expression can occur in any order, which significantly aids in simplifying the expression.