Chemistry Lesson 21 Homework

Algebra 1 Textbook - Chapter 1 - Lesson 21 - Closure Property and Commutative Property

Released - May 23, 2017

In this lesson, we will look at the Closure Property and the Commutative Property.

We are going to explore a little bit of the theoretical side of Algebra here, and this will help to make sense of some of the definitions we are likely to see in our textbooks. While these theories are not usually used directly to solve problems, it is important to understand them in order to know what is going on when we work problems. This understanding will also help us to do things like rearrange problems to make them easier to work with or find steps in our work where we made an error that needs to be fixed.

We need to start by defining what a Real Number is.


DEFINITION

Real Number: Any number on the number line. These include all of the positive numbers, all of the negative numbers, and zero.


Real numbers include all positive and negative integers, fractions, and decimals. `3`, `-14`, `3/4`, `(-11)/(15)`, `3.53343`, `-0,33333`, and `pi` are all examples of real numbers. Simply stated, real numbers are the set of any number you can think of except for imaginary numbers.

Don't worry if you don't know what an imaginary number is right now. Imaginary numbers are very useful, and we will learn about them much later in this course. Imaginary numbers have an `i` in them, so they look something like `5i` or `43i`, for example.


CLOSURE PROPERTY

Closure Property for Addition: If `a` and `b` are real numbers, then `a+b` is unique and real.

Closure Property for Multiplication: If `a` and `b` are real numbers, then `a*b` is unique and real.

This simply means that if we add or multiply any two real numbers together, we will get an answer that is also a real number, and there can be only one answer (it is unique).Textbooks sometimes state this using even more mathematical symbols (such as If `a`, `b` `in RR`, and so on). This looks more complicated than it really is. The following examples will show that this is simply a basic (but important) concept.


EXAMPLE 1

`5+6=11` is real.

Both `5` and `6` are real numbers. When we add them together, we get `11`, which is another real number, and `11` is the only answer we can get by adding `5+6`.

The statement `5+6=11` is real demonstrates the Closure Property for Addition.


EXAMPLE 2

`5*6=30` is real.

Both `5` and `6` are real numbers. When we multiply them together, we get `30`, which is another real number, and `30` is the only answer we can get by multiplying `5*6`.

The statement `5*6=30` is real demonstrates the Closure Property for Multiplication.


COMMUTATIVE PROPERTY

Commutative Property for Addition: If `a` and `b` are real numbers, then `a+b=b+a`

Commutative Property for Multiplication: If `a` and `b` are real numbers, then `a*b=b*a`

This simply states that when we do addition or multiplication, the order of the numbers does not matter.


EXAMPLE 3

`2+10=10+2`

`2` and `10` are real numbers. When we add `2` and `10` together, the order does not matter. `2+10=12`, and `10+2=12`.

The statement `2+10=10+2` demonstrates the Commutative Property for Addition.


EXAMPLE 4

`3*5=5*3`

`3` and `5` are real numbers. When we multiply `3` and `5` together, the order does not matter. `3*5=15`, and `5*3=15`.

The statement `3*5=5*3` demonstrates the Commutative Property for Multiplication.


In this lesson, we looked at the Closure Property and the Commutative Property. In summary, the Closure Property simply states that if we add or multiply any two real numbers together, we will get only one unique answer and that answer will also be a real number. The Commutative Property states that for addition or multiplication of real numbers, the order of the numbers does not matter.

In the next lesson, we will look at the Associative and Equality Properties.

- Мидж торопливо пересказала все, что они обнаружили с Бринкерхоффом. - Вы звонили Стратмору. - Да.

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