Problem 14 - Algebra and Trigonometry

Question

State the property of real numbers being used. \(2(A+B)=2 A+2 B\)

Step-by-Step Solution

Answer

Distributive Property of Multiplication over Addition.

Step 1: Identify the Equation

The equation given is \(2(A+B) = 2A + 2B\). Our goal is to identify which property of real numbers is being demonstrated with this equation.

Step 2: Distributive Property

The equation \(2(A+B) = 2A + 2B\) represents the Distributive Property of Multiplication over Addition. This property states that a number multiplied by a sum is equal to the sum of the number multiplied by each addend individually. Mathematically, it is expressed as \(a(b+c) = ab + ac\).

Key Concepts

Real Numbers

Real numbers include a vast range of numbers that we use in everyday mathematics. They encompass all the numbers we are familiar with: whole numbers, fractions, decimals, and irrational numbers.
Real numbers can be positive or negative, and they also include zero. They are represented on the number line, which helps to visualize how real numbers extend infinitely in both the positive and negative directions.

• Whole Numbers: Numbers without any fractional part. Such as 0, 1, 2, 3, etc.
• Integers: These numbers include all whole numbers and their negative counterparts, like -1, -2, -3, etc.
• Rational Numbers: Numbers that can be expressed as fractions or ratios of integers. Examples include 1/2, 3.5, or -7.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal values, such as π or the square root of 2.

The real number system is crucial because it helps us perform operations like addition, subtraction, multiplication, and division across different types of numbers without switching to a different system of numbers.

Multiplication over Addition

The concept of multiplication over addition is beautifully illustrated by the Distributive Property. This property is fundamental in algebra and is expressed mathematically as \(a(b+c) = ab + ac\).
In simpler terms, it means that if you have a number multiplied by a sum, you can "distribute" that multiplication across each term in the sum.

• For instance, take the equation \(2(A+B)\). The multiplication 2 is distributed to both terms inside the parentheses \(A\) and \(B\), resulting in \(2A + 2B\).
• This method helps simplify equations and is especially useful when solving algebraic expressions or equations.

When used correctly, this property allows for more flexible manipulation of numbers, making complex calculations easier to handle. It's a pivotal tool that streamlines the process of working with algebraic equations and facilitates the understanding of how numbers interact through multiplication and addition.

Algebraic Properties

Algebraic properties like the distributive property help us understand and manipulate mathematical expressions more effectively. They reveal the inherent relationships between numbers and how they can be arranged and calculated.

• Commutative Property: This property assures that the order of numbers doesn't change the result, applicable in both addition (\(a + b = b + a\)) and multiplication (\(a \times b = b \times a\)).
• Associative Property: It states that the grouping of numbers does not affect their sum or product, such as \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: As seen in the problem \(2(A+B) = 2A + 2B\), this property bridges multiplication and addition, highlighting how multiplication can "spread" across terms within an expression.

By understanding these properties, students can solve equations more effectively and improve their proficiency in algebraic manipulation. They provide a framework that not only simplifies calculations but also offers insight into the fundamental nature of arithmetic and algebra.