What Is a Rational Number Definition and Examples

What Is a Rational Number Definition and Examples SAT / ACT Prep Online Guides and Tips Have you heard the term â€Å"rational numbers?† Are you wondering, â€Å"What is a rational number?† If so, you’re in the right place! In this article, we’ll discuss the rational number definition, give rational numbers examples, and offer some tips and tricks for understanding if a number is rational or irrational. What Is A Rational Number? In order to understand what rational numbers are, we first need to cover some basic math definitions: Integers are whole numbers (like 1, 2, 3, and 4) and their negative counterparts (like -1, -2, -3, and -4). Fractions are numbers that are expressed as ratios. A fraction is a part of a whole. Fractions have numerators, which are the numbers on the top of the fraction that show the parts taken from the whole. Fractions also have denominators, which are the numbers on the bottom of the fraction that show how many parts are in the whole. Okay! Now that we know those terms, let’s turn to our original question. What is a rational number? A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers. The denominator in a rational number cannot be zero. Expressed as an equation, a rational number is a number a/b, b≠ 0 where a and b are both integers. This equation shows that all integers, finite decimals, and repeating decimals are rational numbers. In other words, most numbers are rational numbers. Here’s a hint: if you’re working with a number with a long line of different decimals, then your number is irrational! If you’re working with an integer or a number with terminal or repeating decimals (like 1.333333), then your number is rational! Rational Number Examples Now that we know the rational number definition, let’s use that definition to examine some numbers and see if they’re rational or not. Let’s start with the number 6. The number 6 is an integer. It’s also a rational number. Why? Because 6 can also be expressed as 6/1. When expressed as 6, both the numerator and the denominator are integers. The denominator doesn’t equal 0. What about the number -6? -6 can be written as -6/1. Or 6/-1. Either way, -6 is a rational number, because it can be expressed as a fraction where the numerator and denominator are integers and the denominator doesn’t equal 0. What’s an Irrational Number? The opposite of rational numbers are irrational numbers. In simple terms, irrational numbers are real numbers that can’t be written as a simple fraction like 6/1. Take Ï€. Ï€ is a real number. But it’s also an irrational number, because you can’t write Ï€ as a simple fraction: Ï€ = 3.1415926535897932384626433832795 (and counting) There’s no way to write Ï€ as a simple fraction, so it’s irrational. The same goes for √2. The √2 equals 1.4142135623730950...(etc). You can’t make √2 into a simple fraction, so it’s an irrational number. Famous Irrational Numbers There aren’t any famous rational numbers, because the vast majority of numbers are rational. There are a few famous irrational numbers. Here are some ones you might have seen: e: The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527. Ï€: People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 √: Many square roots, cube roots, etc are also irrational numbers. Examples: √3 = 1.7320508075688772935274463415059 (etc) √99 = 9.9498743710661995473447982100121 (etc) Not all square roots are irrational numbers, though! If your square root results in a whole number (like √4 or √9), then you actually are working with a rational number! That’s not the only thing you have to be careful about! Sometimes, multiplying two irrational numbers will result in a rational number. For example, √2 * √2 = 2 2 is a rational number. Main Takeaways Rational numbers are numbers that can be expressed as simple fractions. Irrational numbers are numbers that can’t be expressed as simple fractions. What's Next? Want to know the fastest and easiest ways to convert between Fahrenheit and Celsius? We've got you covered! Check out our guide to the best ways to convert Celsius to Fahrenheit (or vice versa). Are you learning about logarithms and natural logs in math class?We have a guide on all the natural log rules you need to know. Did you know that water has a very special density? Check out our guide to learn what the density of water is and how the density can change.

Deregulating Telecommunications

Deregulating Telecommunications Until the 1980s in the United States, the term telephone company was synonymous with American Telephone Telegraph. ATT controlled nearly all aspects of the telephone business. Its regional subsidiaries, known as Baby Bells, were regulated monopolies, holding exclusive rights to operate in specific areas. The Federal Communications Commission regulated rates on long-distance calls between states, while state regulators had to approve rates for local and in-state long-distance calls. Government regulation was justified on the theory that telephone companies, like electric utilities, were natural monopolies. Competition, which was assumed to require stringing multiple wires across the countryside, was seen as wasteful and inefficient. That thinking changed beginning around the 1970s, as sweeping technological developments promised rapid advances in telecommunications. Independent companies asserted that they could, indeed, compete with ATT. But they said the telephone monopoly effectively shut them out by refusing to allow them to interconnect with its massive network. The First Stage of Deregulation Telecommunications deregulation came in two sweeping stages. In 1984, a court effectively ended ATTs telephone monopoly, forcing the giant to spin off its regional subsidiaries. ATT continued to hold a substantial share of the long-distance telephone business, but vigorous competitors such as MCI Communications and Sprint Communications won some of the business, showing in the process that competition could bring lower prices and improved service. A decade later, pressure grew to break up the Baby Bells monopoly over local telephone service. New technologies- including cable television, cellular (or wireless) service, the Internet, and possibly others- offered alternatives to local telephone companies. But economists said the enormous power of the regional monopolies inhibited the development of these alternatives. In particular, they said, competitors would have no chance of surviving unless they could connect, at least temporarily, to the established companies networks- something the Baby Bells resisted in numerous ways. Telecommunications Act of 1996 In 1996, Congress responded by passing the Telecommunications Act of 1996. The law allowed long-distance telephone companies such as ATT, as well as cable television and other start-up companies, to begin entering the local telephone business. It said the regional monopolies had to allow new competitors to link with their networks. To encourage the regional firms to welcome competition, the law said they could enter the long-distance business once the new competition was established in their domains. At the end of the 1990s, it was still too early to assess the impact of the new law. There were some positive signs. Numerous smaller companies had begun offering local telephone service, especially in urban areas where they could reach large numbers of customers at low cost. The number of cellular telephone subscribers soared. Countless Internet service providers sprung up to link households to the Internet. But there also were developments that Congress had not anticipated or intended. A great number of telephone companies merged, and the Baby Bells mounted numerous barriers to thwart competition. The regional firms, accordingly, were slow to expand into long-distance service. Meanwhile, for some consumers- especially residential telephone users and people in rural areas whose service previously had been subsidized by business and urban customers- deregulation was bringing higher, not lower, prices. This article is adapted from the book Outline of the U.S. Economy by Conte and Carr and has been adapted with permission from the U.S. Department of State.