Problem 10 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The Distributive Property is used.

Step 1: Understanding the Expression

We start with the expression \(2(A + B) = 2A + 2B\). The expression on the left is a product of an expression inside the parentheses and the number 2. On the right, the number 2 multiplies each term inside the parentheses separately.

Step 2: Identify the Property Used

The expression \(2(A + B) = 2A + 2B\) demonstrates the Distributive Property of multiplication over addition. This property states that \( a(b+c) = ab + ac \), meaning you can multiply each addend by the factor outside the parentheses individually and take the sum of the results.

Key Concepts

Real Numbers

Real numbers are a cornerstone in mathematics. They include all the numbers that you can find on the number line. This encompasses both rational numbers, such as integers and fractions, and irrational numbers, like square roots and pi. You can think of them as encompassing the entire set of numbers you could possibly measure or calculate with. Real numbers can be either positive or negative. They include zero. - Rational numbers can be expressed as a fraction where both the numerator and the denominator are integers. - Irrational numbers don't have a precise fraction form and their decimal representation is infinite and non-repeating. It's important to understand real numbers as they apply to various operations like multiplication, addition, and their properties, such as the distributive property. They allow us to perform a wide variety of calculations, providing a structure that supports a vast range of mathematical functions.

Multiplication

Multiplication is one of the basic operations in mathematics. It involves combining equal groups of numbers to find a total amount. For example, if you have 3 groups of 4 apples, you multiply to find that you have 12 apples in total: this is the result of multiplying 3 by 4.Multiplication is closely linked to the distributive property. This property allows you to simplify expressions by breaking them down. For instance, in the expression \(2(A + B)\), you can use the distributive property to express this as \(2A + 2B\). Here, 2 is multiplied by each term within the parenthesis, showcasing how multiplication interacts with addition under this rule.The power of multiplication lies in its ability to make complex calculations simpler and faster, especially when dealing with large numbers or multiple terms in expressions.

Addition

Addition is perhaps the most fundamental mathematical operation. It is the process of bringing two or more numbers together to make a larger number. For instance, adding 3 and 5 gives you 8.In the context of properties like the distributive property, addition plays a vital role. When you see an expression such as \(A + B\), it signifies that A and B are being combined. The distributive property leverages addition by showing how each addend can be multiplied separately when there is a multiplier outside the parentheses. In the equation \(2(A + B) = 2A + 2B\), addition helps to establish the terms that are individually multiplied by 2.Understanding addition, especially in how it interacts with other operations like multiplication, is crucial for mastering algebraic expressions and solving equations.