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# difference between real numbers rational numbers

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Numbers, Rational Numbers, Rational

A number in the form of a ratio a /b, where a and b are integers , and b is not equal to 0, is called a rational number. The rational numbers are a subset of the real numbers, and every rational number can be expressed as a fraction or as a decimal form that either terminates or repeats. Conversely, every decimal expansion that either terminates or repeats represents a rational number. Rational numbers can be written in several different forms using equivalent fractions. For example, . There are an infinite number of ways to write 1, Â¼ or by multiplying both the numerator and denominator by the same nonzero integer. Therefore, there are an infinite number of ways to write every rational number in terms of its equivalent fraction. The following example shows how to find the ratio of integers that represents a repeating decimal. One way to compare two rational numbers is to convert them into a decimal form. Dividing the numerator by the denominator results in the decimal equivalent. If the division has no remainder, then the decimal is called a terminating decimal. For example, Â½ = 0.5, , and . Although some decimals do not terminate, they do repeat because at some point a digit, or group of digits, repeats in a regular fashion. Examples of repeating decimals are â…“ = 0.333â€¦,, and . A bar written over the digits or group of digits that repeat shows that the decimal is repeating: , and . Rational numbers satisfy the following properties. see also Integers; Numbers, Irrational; Numbers, Real; Numbers, Whole. Rafiq Ladhani Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995. Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

Number System, Real Number System, Real

Question:so basically I know that Irrational Numbers can't be written in a fraction and a rational numbers can be written in a fraction. I also know both irrational and rational numbers are real numbers. Is there anything i'm missing? there is a whole section asking whether the number is rational or irrational on my hw and i need to know all the different properties. thanks. :]

Answers:So you are absolutely correct. but as you said rational numbers can be written in a fraction which is not the correct definition. In a fraction we get only positive numbers. But in a rational number we get positive as well as negative integers. for ex; -3/7, -4/5, 4/5 here you should also remember that all fractions are rational numbers but all rational numbers are not fractions. A number which can be expressed in the p/q form where p and q are integers and q is not equal to zero is called a rational number. all others are irrational numbers. and as you said the set of rational and irrational together are called real numbers. hope it is clear to you.

Question:I was reading about them both and they seem similar. Any thoughts on the matter?

Answers:Rational numbers are included in the real numbers. They are a subset of the real numbers.

Question:What is the difference between rational numbers and irrational numbers? Would v2 (that's "root 2". lol) be a rational number? What about the fraction 8/9? Also, what is a "real number"? Would the numbers I posted above be real numbers as well?

Answers:it's simple: rational numbers are real numbers (read: bottom) that can be expressed through fractions, whole numbers, decimals, and mixed numbers. They can be written as a ratio of two integers. irrational are the leftovers...they can't be expressed as a rational number. an easy way to remember the difference is that rational numbers are "nameable"...for instance you can "name" the following: 2, 8/9, 6.2, -5, sqrt(4), etc. their "names" are: two, eight-ninths, six point two, negative five, the square root of four is +/- two...respectively. on the other hand, irrational numbers aren't "nameable"...such as: sqrt(2), sqrt(3), sqrt(7), etc. they don't have "names" other than just saying what they express: the square root of two, the square root of three, the square root of seven...respectively. See the difference? a real number is any number that exists on the number line...real numbers are composed of rational and irrational numbers. Thus, the numbers you gave are both real numbers, sqrt(2) being an irrational number, and 8/9 being a rational number. if you're still shady, just email. =]

Question:

Answers:A real number is rational if it is given by the ratio of 2 integers. For example, 3/7, 2/3, 1/8, etc. A number is irrational if it cannot be given by the ratio of 2 integers. For example, sqrt(2), 7^(1/3), pi. Rational numbers always have a finite or infinite but periodic decimal representation. Irrational numbers always have an infinite and non periodic decimal representation.