| 单词 | rational number |
| 例句 | If we were to throw a dart at the number line, it would never hit a rational number. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Cantor’s theory would show that Kronecker's precious integers—and even the rational numbers—were nothing at all. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Let’s do the same thing with the rational numbers—but this time our carpets are sets of numbers. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Yet even those tiny carpets, all of which can fit in the span of an atom, cover all of the rational numbers. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Using this sort of carpet to cover the rational numbers has some very odd consequences, as Cantor soon showed, thanks to his seating chart. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z So we can cover the infinite cohorts of rational numbers in the number line with a set of carpets, and the total size of the carpets is 2. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z This means that the rational numbers take up less than two units of space. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z It was built upon rational numbers and implied the existence of God. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Go on and on to infinity; since every rational number is on the seating chart, every rational number will eventually be covered by a carpet. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z But Cantor’s hierarchy of infinities would tell a different tale: it would show just how little space the rational numbers take up on the number line. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z If we start with a carpet the size of half an atom, we can cover all the rational numbers on the number line with carpets that, in total, take up less room than an atom. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Even though there are rational numbers everywhere on the number line, they take up no space at all. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z The real numbers are a bigger infinity than the rational numbers. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z That seating chart accounts for all the rational numbers—it assigns each of them a seat—so we can count them off one by one, in order, based on their seat number. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Even the rational numbers—the set of numbers that can be written as a/b for integers a and b—were countable. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z Take the first rational number and imagine it on the number line. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z And the box “What Are Imaginary Numbers?” should have said that rational numbers include the integers, not that “rational numbers are the integers.” Readers Respond to the February 2023 Issue 2023-05-28T04:00:00Z Even such bonkers-looking numbers, however, together with all the rational numbers, make up only a tiny fraction of the real numbers, or numbers that can appear along a number line. These Are the Most Bizarre Numbers in the Universe 2023-05-23T04:00:00Z If the two new arrangements contained repeating patterns, the length of their translation vectors should have been related to each other—specifically, their ratio should have been a rational number. Newfound Mathematical ‘Einstein’ Shape Creates a Never-Repeating Pattern 2023-04-10T04:00:00Z TRY IT #1 Write each of the following as a rational number. College Algebra with Corequisite Support 2020-09-23T00:00:00Z Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. High School Statistics 2020-03-27T00:00:00Z So the list of rational numbers is −7, 14, 8, 5.9, and − 64. Intermediate Algebra 2020-05-06T00:00:00Z When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator. Elementary Algebra 2020-04-22T00:00:00Z Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. Calculus, Volume 1 2016-03-30T00:00:00Z Because they are fractions, any rational number can also be expressed in decimal form. Algebra and Trigonometry 2015-02-13T00:00:00Z The conjecture predicts that the corresponding field will be “the smallest field that you get by taking sums, products and rational number multiples of these roots,” Arthur says. The Evolving Quest for a Grand Unified Theory of Mathematics 2022-03-21T04:00:00Z In your own words, explain the difference between a rational number and an irrational number. Intermediate Algebra 2020-05-06T00:00:00Z If we square an irrational square root, we get a rational number. Elementary Algebra 2020-04-22T00:00:00Z In Table 1.11, we list some rational numbers approaching 2, and the values of 2 x for each rational number x are presented as well. Calculus, Volume 1 2016-03-30T00:00:00Z The circumference of the quarter would be the diameter multiplied by Is the circumference of a quarter a whole number, a rational number, or an irrational number? Algebra and Trigonometry 2015-02-13T00:00:00Z Poonen has worked to show that such a general method for finding solutions in rational numbers is impossible. Gnarly, Centuries-Old Mathematical Quandaries Get New Solutions 2021-10-14T04:00:00Z We introduced rational numbers, which are just fractions where the numerators and denominators are integers. Intermediate Algebra 2020-05-06T00:00:00Z We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero. Elementary Algebra 2020-04-22T00:00:00Z This extension will ultimately allow us to differentiate xq, where q is any rational number. Calculus, Volume 1 2016-03-30T00:00:00Z But first we need a pool of rational numbers to test. Algebra and Trigonometry 2015-02-13T00:00:00Z For example, the rational numbers within the real line are a null set even though there are infinitely many of them. A Deep Math Dive into Why Some Infinities Are Bigger Than Others 2021-08-16T04:00:00Z When they needed the idea of parts of a whole they added fractions and got the rational numbers. Intermediate Algebra 2020-05-06T00:00:00Z In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers. Elementary Algebra 2020-04-22T00:00:00Z Rational numbers are dense because any interval, no matter how small its length, contains at least one rational number. How Quickly Can You Fill Up a Circle? 2019-09-08T04:00:00Z Recall that a rational number is the ratio of two numbers, such as or A rational expression is the ratio, or quotient, of two polynomials. Algebra and Trigonometry 2015-02-13T00:00:00Z Even though most numbers—in a sense, 100% of them—are irrational, humans are strangely obsessed with rational numbers. I Can Has Numberz? 2019-05-05T04:00:00Z All of our exponential functions have had either an integer or a rational number as the base. Intermediate Algebra 2020-05-06T00:00:00Z A rational number is a number that can be written as a ratio of two integers. Elementary Algebra 2020-04-22T00:00:00Z Adding a rational number over and over, on the other hand, is not dense because eventually you will be repeating the same number. How Quickly Can You Fill Up a Circle? 2019-09-08T04:00:00Z We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1. Algebra and Trigonometry 2015-02-13T00:00:00Z Other sets of numbers, such as prime numbers and rational numbers, are also countable because with a little ingenuity they can be listed. The Unknowability of the Number Line 2019-01-12T05:00:00Z You can extend the p-adic absolute value and distance functions to all rational numbers. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z We have seen that every integer is a rational number, since a = a for any integer, a. Elementary Algebra 2020-04-22T00:00:00Z “I’m hopeful that we’ll see a more rational number going forward and if it is more rational, then I think we’ll be able meet the standard,” he said. Major automakers urge Trump administration: don't ditch NAFTA 2018-01-15T05:00:00Z Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational. Algebra and Trigonometry 2015-02-13T00:00:00Z If the entire number line is uncountable, and the rational numbers are countable, then it stands to reason that the irrational numbers must be uncountable. The Unknowability of the Number Line 2019-01-12T05:00:00Z There are sequences of rational numbers that do not end up at a rational number in the p-adic world. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z Given the numbers −7, 14, 8, 5, 5.9, − 64, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers. Elementary Algebra 2020-04-22T00:00:00Z “I’m hopeful that we’ll see a more rational number going forward and if it is more rational, then I think we’ll be able meet the standard,” he said. Major automakers urge Trump administration: don't ditch NAFTA 2018-01-15T05:00:00Z The rational numbers and irrational numbers make up the set of real numbers. Algebra and Trigonometry 2015-02-13T00:00:00Z But in the context of the so-called rational numbers, which include fractions, 5 loses its prime identity and gains versatility, able to be divided into ever tinier slivers, like a cake at a dieters’ convention. Eugenia Cheng Makes Math a Piece of Cake 2016-05-02T04:00:00Z After figuring out how to represent all rational numbers p-adically and how to do arithmetic with them, we find that something’s missing. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z The numbers we would have form the set of rational numbers. Elementary Algebra 2020-04-22T00:00:00Z But a growing body of scientific evidence suggests that we are not the consistent, rational number crunchers that we suppose. Irrationality Continues to Plague the Justice System [Excerpt] 2015-06-19T04:00:00Z The set of rational numbers, in turn, fills a void left by the set of integers. Algebra and Trigonometry 2015-02-13T00:00:00Z The golden ratio is the irrational number that is hardest to approximate with rational numbers. Strange Stars Pulsate According to the "Golden Ratio" 2015-02-09T05:00:00Z We can find rational numbers that are closer and closer to the length of the circumference of a circle with diameter 1, but the number π is not rational. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z To decide if an integer is a rational number, we try to write it as a ratio of two integers. Elementary Algebra 2020-04-22T00:00:00Z The class is then promptly divided into four small groups, each guided by a BETA—a Blue Engine Teaching Assistant, there to help with factoring and rational number expressions. Reclaiming the American Dream V: In the Bronx, A (Blue) Engine of Opportunity 2014-10-21T04:00:00Z The set of real numbers fills a void left by the set of rational numbers. Algebra and Trigonometry 2015-02-13T00:00:00Z But some irrational numbers are easy to approximate using rational numbers whereas others are hard. Strange Stars Pulsate According to the "Golden Ratio" 2015-02-09T05:00:00Z Using the usual distance between numbers, we also feel like the rational numbers are incomplete. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z Since any integer can be written as the ratio of two integers, all integers are rational numbers! Elementary Algebra 2020-04-22T00:00:00Z And the integers and rational numbers—the easiest real numbers to think about—aren’t normal. What Is the Funniest Number? 2013-08-05T22:15:02.703Z If it’s finite, it’s not too hard to see that the continued fraction represents a rational number. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z For instance: Have we proved that the rational numbers are countably infinite? An eternity of infinities: the power and beauty of mathematics 2013-01-23T00:45:00.190Z We can think of sequences of rational numbers that, when squared, get closer and closer to 2, but there is no rational number whose square is 2. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z We have also seen that every fraction is a rational number. Elementary Algebra 2020-04-22T00:00:00Z When a and b are given rational numbers, c and d may be chosen in an infinite number of ways as rational numbers, so that d − ad + bc shall be any assigned rational number. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z A summer fling might be a rational number with a relatively small denominator, and an on-again, off-again romance might be a square root. Romance, Continued Fractions, and You 2013-02-14T15:15:08.637Z They were also the first to study seriously the incommensurable number, and with this study the treatment of proportion from the standpoint of rational numbers lost its scientific position with respect to geometry. The Teaching of Geometry 2011-10-12T02:00:52.133Z One of the motivating questions in that field is when there are whole number or rational number solutions to polynomial equations. The Numbers Behind a Fields Medalist's Math 2018-09-03T04:00:00Z We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Elementary Algebra 2020-04-22T00:00:00Z Moreover, the totality of additions with the identical operation, i.e. the totality of operations of the form x′ = x + b, where b may be any rational number or zero, obviously constitutes a group. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z The aggregates of all rational numbers and of all real numbers and of points on a line are other examples of infinite aggregates. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z An interesting theory suggested by the proposition is that of computing the sides of right triangles so that they shall be represented by rational numbers. The Teaching of Geometry 2011-10-12T02:00:52.133Z The Pythagoreans gave general solutions in rational numbers of the equations x²+y²=z² and 2x²-y²=±1, which are indeterminate equations of the second degree. The Legacy of Greece Essays By: Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D'arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield When we put together the rational numbers and the irrational numbers, we get the set of real numbers. Elementary Algebra 2020-04-22T00:00:00Z The totality of multiplications, with the identical operation, i.e. all operations of the form x′ = ax, where a is any rational number other than zero, again obviously constitutes a group. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z Boy: So we define the rational numbers as numbers made from the division into ratios of whole numbers, whether those whole numbers are even or odd. The Second Story of Meno; a continuation of Socrates' dialogue with Meno in which the boy proves root 2 is irrational A rational number can be written as the ratio of two integers. Elementary Algebra 2020-04-22T00:00:00Z In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. Elementary Algebra 2020-04-22T00:00:00Z In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers. Elementary Algebra 2020-04-22T00:00:00Z When d is a given rational number, the set constitutes a subgroup which is conjugate to the group of multiplications. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z The totality of rational numbers, including zero, forms a set of objects which are interchanged among themselves by all operations of the group. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z The group is therefore doubly transitive in respect of the set of rational numbers. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z The subgroups, each of which leaves a single rational number unchanged, therefore form a single conjugate set. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z |
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