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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

The question "How many?" prompted early civilizations to make marks to record the answers. The words and signs used to record how many were almost surely related to our body parts: two eyes, five fingers on one hand, twenty fingers and toes. For instance, the word "digit," which we use for the symbols that make up all our numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), is the Latin word for finger. These first numbers are now called the set of counting numbers: {1, 2, 3,â€¦}, and sometimes this set is called the natural numbers. Notice that the counting numbers do not include 0. Whereas some early cultures, including the Egyptians, the Chinese, and the Mayans, understood the concept of zero, the digit 0 did not appear until some time after the other nine digits. In the earliest development of counting, many languages used "one, two, many," so that the word for three may have meant simply "much." People everywhere apparently developed counting systems based on repeating some group, as in counting "one group of two, one group of two and one, two groups of two." We know that a scribe in Egypt used number signs to record taxes as early as 2500 b.c.e. Hundreds of unbaked clay tablets have been found that show that the Babylonians, in the region we know today as Iraq, were using marks for one and for ten in the 1,700 years before the birth of Christ. These tablets show that the idea of place value was understood and used to write numbers. Number signs were needed not only to count sheep or grain but also to keep track of time. Many civilizations developed complex mathematical systems for astronomical calculations and for recording the calendar of full moons and cycles of the Sun. These earliest records included fractions, or rational numbers, as well as whole numbers, and used place value in ways similar to the way decimal fractions are written today. In a manuscript, possibly from the sixth century, fractions were written with the numerator above and the denominator below, but without the division line between them. The bar used for writing fractions was apparently introduced by the Arabs around 1000 c.e. Early forms of the Arabic-Hindu numerals, including 0, appeared sometime between 400 c.e. and 850 c.e., though recent evidence suggests that 0 may have been invented as early as 330 b.c.e. The zero sign began as a dot. It is possible that the late development of 0 was because people did not see zero as a meaningful solution when they were doing practical problems. About 850 c.e., a mathematician writing in India stated that 0 was the identity element for addition, although he thought that division by 0 also resulted in a number identical to the original. Some 300 years later, another Hindu mathematician explained that division by 0 resulted in infinity. Number rods were used by the Chinese as a computational aid by 500 b.c.e. The Koreans continued to use number rods after the Chinese and the Japanese had replaced the counting rods with beads in the form of an abacus. Red rods represented positive numbers, and black rods represented negative numbers. The book Arithmetica, by Diophantus (c. 250 c.e.), calls an equation such as 4x + 20 = 4 "absurd" because it would lead to x âˆ’4. Negative numbers are mentioned around 628 c.e. in the work of an Indian mathematician, and later they appear in all the Hindu math writings. Leonardo Pisano Fibonacci, writing in 1202, paid no attention to negative numbers. It was not until the Renaissance that mathematics writers began paying attention to negative numbers. The idea of letting a variable, such as a or x, represent a number that could be either positive or negative was developed around 1659. The negative sign as we know it began to be used around 1550, along with the words "minus" and "negative" to indicate these numbers. The idea of square roots, which leads to irrational numbers such as , apparently grew from the work of the Pythagoreans with right triangles. Around 425 b.c.e., Greeks knew that the square roots of 3, 5, 6, and 7 could not easily be measured out with whole numbers. Euclid, around 300 b.c.e. classified such square roots as irrational; that is, they cannot be expressed as the ratio of two whole numbers. The history of the development of human knowledge of the real numbers is not clearly linear. Different people in widely separated places were thinking and writing about mathematics and using a variety of words and notations to describe their conclusions. The development of numbers that are not realâ€”that is, of numbers that do not lie on what we today call the real number lineâ€”began around 2,000 years ago. The square root of a negative number, which leads to the development of the complex number system, appears in a work by Heron of Alexandria around 50 c.e. He and other Greeks recognized the problem, and Indian mathematicians around 850 stated that a negative quantity has no square root. Much later, in Italy, after the invention of printing, these roots were called "minus roots." In 1673, Wallis said that the square root of a negative number is no more impossible than negative numbers themselves, and it was he who suggested drawing a second number line perpendicular to the real number line and using this as the imaginary axis. see also Calendar, Numbers in the; Integers; Mathematics, Very Old; Number Sets; Numbers and Writing; Numbers, Complex; Numbers, Irrational; Numbers, Rational; Numbers, Real; Numbers, Whole; Radical Sign; Zero. Lucia McKay Eves, Howard. An Introduction to the History of Mathematics. New York: Holt, Rinehart and Winston, 1964. Hogben, Lancelot. Mathematics in the Making. London: Crescent Books, 1960. â€”â€”. The Wonderful World of Mathematics. New York: Garden City Books, 1995. Steen, Lynn Arthur, ed. On the Shoulders of Giants, New Approaches to Numeracy. Washington, D.C.: National Academy Press, 1990.

From Yahoo Answers

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.

From Youtube

Real Numbers: Rational and Irrational Numbers are Real Numbers :