Hi

Do decimals such as 2.718 represent rational numbers or irrational numbers. Explain.

Do repeating decimals such as 2.3333 . . . represent rational numbers or irrational numbers? Explain.

So I look back in my textbook and find

rational numbers
(e.g., 4/5, -2/3, 7.31, -5, square root 9, o.3333 . . .)
Can be expressed exactly as a ratio of two integers.

Irational Nmbers
(e.g., square root five, -3 root 11, Pi)
Cannot be expressed exactly as a ratio of two integers, but are real numbers.

Intergers
(e.g., 2, -17, 2001, 0)
Whole numbers and their opposites.

Ok so I am confused. A rational number can be expressed exactly as a ratio of two integers. Well sense numbers with decimals aren't integers sense according to my book an integer has to be a whole number. So dosen't all numbers with decimals have to be irrational sense they are not integers?

What's the difference between natural numbers and counting numbers? Are they the same thing?

Lastly in my textbook it litsts 0 as as an example for a digit. 0 also has it's own group of being neither positive nor negative nor no value. So if I were to see something on a test that told me tell what set of numbers given numbers were and saw zero... would I say it was zero or a digit?

thanks

Decimals such as 2.718 are examples of irrational numbers. An irrational number cannot be expressed exactly as a ratio of two integers. In the case of 2.718, it is the mathematical constant known as Euler's number and is irrational.

Repeating decimals like 2.3333... are rational numbers. These numbers can be expressed as a ratio of two integers. In the case of 2.3333..., it can be written as the fraction 7/3.

Now, let's address your confusion about rational and irrational numbers. Integers are indeed whole numbers and their negatives. Rational numbers can be expressed as a ratio of two integers, but they can also include numbers with decimal representations that eventually terminate or repeat. Irrational numbers, on the other hand, cannot be expressed as a ratio of two integers and have decimal representations that do not terminate or repeat.

Regarding natural numbers and counting numbers, they are essentially the same thing. Both terms refer to the set of positive integers starting from 1 and going indefinitely.

Lastly, if you come across a question that asks you to identify the set of numbers for a given set of digits and zero is included, you would classify zero as an integer because it belongs to the set of whole numbers and their negatives. The term "digit" simply refers to the symbols used to represent numbers.

Decimals, such as 2.718, can represent both rational and irrational numbers, depending on their properties.

Let's first clarify the concept of rational numbers. As your textbook mentions, rational numbers can be expressed exactly as a ratio of two integers. This means that when you write a rational number as a fraction, both the numerator and denominator are integers.

For example, the decimal 7.31 can be written as the fraction 731/100, where both 731 and 100 are integers. So, 7.31 is a rational number.

Now, let's discuss the decimal 2.718. This number is special and is equal to the mathematical constant e, approximately 2.718281828459045. It cannot be expressed as a ratio of two integers. In fact, it is an irrational number. Irrational numbers cannot be expressed exactly as a ratio of two integers but can still be represented as a decimal or a root.

So, to answer your first question, the decimal 2.718 represents an irrational number.

Moving on to your second question about repeating decimals, such as 2.3333... By definition, repeating decimals are decimals that have a repeating pattern. These decimals can always be expressed as a fraction, making them rational numbers.

In the case of 2.3333..., it can be written as the fraction 7/3. Here, the digit "3" repeats infinitely, which is denoted by the ellipsis (...). So, 2.3333... is a rational number.

Regarding the difference between natural numbers and counting numbers, they are often used interchangeably, depending on the context. However, some definitions make a distinction between the two.

Natural numbers usually refer to the set of positive integers (1, 2, 3, 4, ...), including zero in some cases. Counting numbers typically refer to the set of positive integers only (1, 2, 3, 4, ...), excluding zero.

In most cases, if the context is not clear, it is safe to assume that natural numbers include zero, but it's always best to clarify with your teacher or textbook.

Lastly, regarding the number zero, it is a unique element in mathematics. It can be considered both a digit and a member of the set of integers. As a digit, it represents the absence of quantity (e.g., in the number 10, the zero indicates there are no units).

If you encounter a question asking about the set of numbers a given number belongs to and zero is one of the options, you would classify it as an integer since it belongs to the set of integers. However, it's important to note that zero is not positive, negative, or non-existent; it is simply its own entity within the number system.

Please learn to spell A L G E B R A.