9.0 Rational Or Irrational

[Idara Viengxay]

Fractions can also be written as decimals. For example, \(\frac13100\) is equal to 0.13 because the 3 is in the hundredths decimal place and the 1 is in the tenths decimal place. This is an example of a terminating decimal.

It is important to note that not all decimals are repeating. Some decimals have an infinite number of non-repeating digits. These types of real numbers cannot be expressed as a ratio of integers and are therefore classified as irrational.

We can keep it simple and do 3.25, 3.5, and 3.75, but in reality any terminating decimal like 3.58 or 3.987 would work in this scenario. Remember, any number between 3 and 4 that can be written as a fraction would also be correct.

Yes, most negative numbers are rational. A rational number is any number that can be written as a fraction. These include whole numbers, fractions, decimals that end, and decimals that repeat. Positive and negative do not affect rationality.

No, not all rational numbers are whole numbers. Rational numbers include all numbers that end or repeat. A whole number is any number without a fractional part that is greater than or equal to zero.
Ex. 2.7 is a rational number but not a whole number.

The difference between rational and irrational numbers is that a rational number can be represented as an exact fraction and an irrational number cannot. A rational number includes any whole number, fraction, or decimal that ends or repeats. An irrational number is any number that cannot be turned into a fraction, so any number that does not fit the definition of a rational number.

The correct answer is \(\sqrt3\). Square roots of non-perfect squares are not rational because they are equal to a never-ending decimal number, which means it is a number that cannot be turned into a fraction.

The primary difference between rational numbers and irrational numbers is whether the numbers can be written as fractions. In order to determine whether a number is rational or irrational, you must check to see if the number can be written as a fraction.

I think Thrall has got this right. Endless discussions of why this or that initiative or attempt to mediate failed are shown to be superfluous. We can stop wondering why the whole process of negotiations, beginning in the late 1980s, has remained so barren. Was it because Ehud Barak was not very courteous to Yasser Arafat at Camp David in 2000? Or because Ehud Olmert was burdened by scandal and political crisis when he finally made an offer to Palestinian President Mahmoud Abbas in 2008? It has been clear for many years that the very notion of peace negotiations between the two parties has been little more than a device to perpetuate, not to end, the occupation. As Thrall writes:

The United States has consistently sheltered Israel from accountability for its policies in the West Bank by putting up a faade of opposition to settlements that in practice is a bulwark against more significant pressure to dismantle them.

Steinberg is no less interested in Palestinian extremists than in pragmatic centrists, if such a word is appropriate in a polity so weakened and diffuse. He never underestimates the power of Hamas and the militant factions. Again and again he shows the diabolical interplay between such groups and the dominant Israeli policy of strengthening the occupation:

Two months earlier, on January 7, the same settlers from Chavat Maon violently attacked a group of Israeli peace activists who were accompanying Palestinian farmers seeking to plow a field. I was there with another party of activists, a little farther down the hill, and I witnessed the arrival of the wounded in Twaneh: one hit by a rock on the head, two others badly beaten, still more with contusions, and one with a smashed camera.

Children from the Twaneh area are at constant risk of being attacked by settlers on their way to school in the village; the daughter of a friend of mine, Ali from Tuba, nearly lost an eye in such an attack. The army has been forced to provide a military escort to take them to and from school, but even that is not always enough; there have been occasions when the soldiers stood idly by while settlers beat the Palestinian children with clubs and metal chains.

In the northern Jordan Valley, Bedouin shepherds from a tiny place called al-Hammeh are subject to continuous attacks by settlers from a new illegal settlement that sits on the al-Hammeh land; these settlers have murdered Bedouin sheep, threatened the shepherds with guns, beaten them savagely, invaded their tents, and in general done whatever they can to make their lives miserable.1 At nearby al-Auja, on April 21, a gang of masked Israeli settlers from Habaladim, an illegal West Bank outpost, used clubs and rocks to attack a group of Palestinian shepherds and more than a dozen Israeli activists who were there to protect them. The result: one activist with an open head wound, another with a broken arm, and several others badly bruised.

The settlers themselves, however obnoxious, bear only a portion of the blame for the atrocities they commit. They carry out the policies of the Israeli government, in effect maintaining a useful, steady level of state terror directed against a large civilian population. None of this can be justified by rational argument. All of it stains the character of the state and has, in my experience, horrific effects on the minds and hearts of young soldiers who have to carry out the orders they are given. A few unusually aware and conscientious ones have had the courage to speak out; as always in such situations, most people just go along.2

There exist other templates for some sort of resolution. The most interesting and creative is probably the Two States One Homeland proposal by Meron Rapoport, Awni al-Mashni, and the group of Palestinians and Israelis they have gathered around them. They envision two states within a single geographical space and a movement toward simultaneous sharing and separation. The blueprint speaks of two independent polities with Jerusalem as their capital; freedom of movement and even freedom to settle on both sides of the border, subject to agreement on the number of citizens of each state who will become permanent residents of the other; a Joint Court for Human Rights, a Joint Security Council, and other common institutions functioning alongside the institutional structures of each state.3

When you hear the words "rational" and "irrational," they might bring to mind the relentlessly analytical Spock in "Star Trek." If you're a mathematician, however, you probably think of ratios between integers versus square roots.

In the realm of mathematics, where words sometimes have specific meanings that are very different from everyday usage, the difference between rational and irrational numbers doesn't have anything to do with emotions. Since there are infinite irrational numbers, you'd do well to gain a basic understanding of them.

"In remembering the difference between rational and irrational numbers, think one word: ratio," explains Eric D. Kolaczyk. He's a professor in the department of mathematics and statistics at Boston University and the director of the university's Rafik B. Hariri Institute for Computing and Computational Science & Engineering.

Irrational numbers, in contrast to rational numbers, are pretty complicated. As Wolfram MathWorld explains, they cannot be expressed by fractions, and when you try to write them as a number with a decimal point, the digits just keep going on and on, without ever stopping or repeating a pattern.

Since Babylonian mathematicians attempted to calculate pi nearly 4,000 years ago, successive generations of mathematicians have kept plugging away, coming up with longer and longer strings of the decimal expansion with non-repeating patterns.

One of the most conspicuous examples is the square root of 2, which works out to 1.414 plus an endless string of non-repeating digits. That value corresponds to the length of the diagonal within a square, as first described by the ancient Greeks in the Pythagorean theorem.

"We do indeed typically use 'rational' to mean something more like based on reason or similar," Kolaczyk says. "Its use in mathematics seems to have cropped up as early as the 1200s in British sources (per the Oxford English Dictionary). If you trace both 'rational' and 'ratio' back to their Latin roots, you find that in both cases the root is about 'reasoning,' broadly speaking."

While language probably dates back to around the origin of the human species, numbers came along much later, explains Mark Zegarelli, a math tutor and author who has written 10 books in the "For Dummies" series. Hunter-gatherers, he says, probably didn't need much numerical precision, other than the ability to roughly estimate and compare quantities.

"Suppose you build a house with a roof for which the rise is the same length as the run from the base at its highest point," Kolaczyk says. "How long is the stretch of roof surface itself from top to outer edge? Always a factor of the square root of 2 of the rise (run). And that's an irrational number as well."

In the technologically advanced 21st century, irrational numbers continue to play a crucial role, according to Carrie Manore. She's a scientist and a mathematician in the Information Systems and Modeling Group at Los Alamos National Laboratory.

"Pi is an obvious first irrational number to talk about," Manore says via email. "We need it to determine area and circumference of circles. It's critical to computing angles, and angles are critical to navigation, building, surveying, engineering and more. Radio frequency communication is dependent on sines and cosines which involve pi."

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to reason that was also proving that the number in question was irrational.

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