Problem 4 - Algebra and Trigonometry

Question

\(3-10=\) State the property of real numbers being used. $$ 2(3+5)=(3+5) 2 $$

Step-by-Step Solution

Answer

1. The subtraction gives -7. 2. The Commutative Property of Multiplication is used.

Step 1: Identify the Mathematical Operation

In this exercise, we first need to compute the subtraction given by the expression \(3-10\). This is a basic arithmetic operation where you subtract 10 from 3.

Step 2: Perform the Subtraction

Now, perform the subtraction \(3 - 10\). Since 10 is greater than 3, the result will be a negative number. Thus, \(3 - 10 = -7\).

Step 3: Understand the Property of Real Numbers

For the second part of the exercise, we analyze the equation \(2(3+5)=(3+5)2\). This equation demonstrates the Commutative Property of Multiplication. The Commutative Property states that for any numbers \(a\) and \(b\), \(a \cdot b = b \cdot a\). Here, \(2\) and \((3+5)\) switch places.

Key Concepts

Real Numbers

Real numbers are all the numbers that can be found on the number line. This includes a variety of different kinds of numbers like:

• Whole numbers
• Integers (which are positive and negative whole numbers)
• Fractions
• Decimals
• Irrational numbers like \( \pi \) and \( \sqrt{2} \)

Real numbers can be positive, negative, or zero. They are used in measuring quantities and they form the building blocks for many mathematical operations. In the context of our exercise, the numbers involved like 3 and 10 are examples of real numbers. Understanding real numbers is essential because they serve as the basis for more complex mathematical concepts.

Basic Arithmetic Operations

Arithmetic operations form the foundation of most calculations in mathematics. The four basic types are:

• Addition
• Subtraction
• Multiplication
• Division

In any arithmetic operation, especially with real numbers, these operations follow certain properties. For instance, when we look at the exercise, subtraction is key to finding the result of \(3 - 10\). Similarly, when analyzing the equation \(2(3+5) = (3+5)2\), we're dealing with multiplication and its properties, specifically the commutative property. Understanding these operations helps in solving simple equations and lays the groundwork for higher-level math.

Subtraction

Subtraction is one of the basic arithmetic operations. It involves taking one number away from another. In simpler terms, if you have 3 apples and you subtract 10 apples, you end up missing 7 apples, which in numeric terms is \(3 - 10 = -7\).
Subtraction has certain characteristics:

• It's not commutative, meaning that \(a - b eq b - a\) in most cases.
• It often results in a negative number if the subtrahend (the number being subtracted) is larger than the minuend (the number it is taken from).

Learning to subtract correctly is crucial, as this operation is widely used in different aspects of real-life scenarios, like balancing a checkbook or calculating change.