Problem 11 - Algebra and Trigonometry

Question

State the property of real numbers being used. \(7+10=10+7\)

Step-by-Step Solution

Answer

This is the commutative property of addition.

Step 1: Identify the Property

Look at the equation provided: \(7 + 10 = 10 + 7\). Recognize that the problem involves the addition of two numbers, and the order in which they are added is changed.

Step 2: Recall the Commutative Property

Remember that the commutative property of addition states that changing the order of the numbers you are adding does not change the sum. This can be written as \(a + b = b + a\).

Step 3: Apply the Commutative Property

In this problem, \(7 + 10 = 10 + 7\) is an example of the commutative property of addition in action, as the order of the addends has been switched, but the sum remains the same.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics that include a vast set of numbers. They encompass many different kinds of numbers, such as:

• Integers: These include positive numbers, negative numbers, and zero. Examples are -3, 0, and 7.
• Rational numbers: These are numbers that can be expressed as a quotient of two integers, like -1/2 or 3/4.
• Irrational numbers: These cannot be precisely expressed as a fraction of two integers, for example, the square root of 2 or pi (\(\pi\)).

Real numbers can be found on the number line, where each point corresponds to a real number. They are useful in every day calculations and measurements like distances and time spans.
They are also integral to the study of mathematical properties, including those relevant to operations like addition and multiplication.

Addition

Addition is one of the most basic operations in mathematics. It involves calculating the total of two or more numbers signified by the '+' symbol.
The operation is important because it lays the foundation for more complex processes and problem solving.
Here are some key points about addition:

• Adding numbers: When you add numbers, you combine them into a single sum. For instance, adding 7 and 10 gives 17.
• Identifying parts: In an addition equation, such as 7 + 10, '7' and '10' are called "addends," and the result is the "sum."
• Operations like addition are key aspects of arithmetic, the branch of math dealing with numbers and their operations.

The beauty of addition lies in its stability and practicality, especially when combined with mathematical properties like the commutative property that makes the order of addends interchangeable without affecting the sum.

Mathematical Properties

Mathematical properties refer to the rules governing operations on numbers that remain consistent across varying calculations.
One such property that is fundamental to understanding addition with real numbers is the commutative property.

• Commutative Property of Addition: This matters a lot in simplifying equations because it states that the order of addition does not alter the sum. For example, \(7 + 10 = 10 + 7\).
• Properties like the associative and distributive properties also influence calculations and simplify problem-solving processes by allowing for the restructuring of equations.

Understanding these properties helps in recognizing patterns and solving mathematical problems more easily.Always remember, mathematical properties are powerful tools that make calculations predictable and systematic.