The quotient of two nonzero integers is the definition of a rational number. There are nonzero numbers other than integers (imaginary, rational non-integers) that the quotient of would not be a rational number. If the two nonzero numbers are rational themselves, then the quotient will be rational. (For example, 4 divided by 2 is 2: all of those numbers are rational).
Because 1. Positive integers are greater than negative integers, and 2. Division by a positive number preserves the order.
A rational number is a number than can be written p/q with p and q integers Any integers can be written this was with q=1
A rational number is one that can be expressed as the ratio of two integers. There are an infinite number of both rational and irrational numbers, but there are more irrational numbers than rational ones... infinitely more, in fact.
Prime numbers have only two factors. Composite numbers have more than two.
A rational number can be expressed as a fraction a/b, where a and b are integers. This includes the integers themselves, that can be expressed as (number) / 1. An irrational number is one that can't be exactly expressed in this way. This includes the square roots of any integer that is not a perfect square; pi; and e - and a host of other numbers. There are more rational than irrational numbers (rational numbers are a countable infinity", and irrational numbers an "uncountable infinity"), but we usually deal mainly with rational numbers - or use rational approximations for the numbers we deal with.
1. No.The Natural numbers are the positive integers (sometimes the non-negative integers).Rational numbers are numbers that can be expressed as the quotient of two integers (positive or negative). All Natural numbers are in the set of Rational numbers. 2. No. Natural numbers are usually defined as integers greater than zero. A Rational number is then defined simply as a number that can be expressed as an integer divided by a natural number. (This definition includes all rational numbers, but excludes division by zero.)
If the divisor was larger than one of the numbers, it couldn't be a divisor. There is no factor of a number that is larger than the number.
The set of irrational numbers is larger than the set of rational numbers, as proved by Cantor: The set of rational numbers is "countable", meaning there is a one-to-one correspondence between the natural numbers and the rational numbers. You can put them in a sequence, in such a way that every rational number will eventually appear in the sequence. The set of irrational numbers is uncountable, this means that no such sequence is possible. All rational and irrationals (ie real numbers) are a subset of complex numbers. Complex numbers, in turn, are part of a larger group, and so on.
There are infinitely many rational numbers and irrational numbers but the cardinality of irrationals is larger by an order of magnitude.If the cardinality of the countably infinite rational numbers is represented by a, then the cardinality of irrationals is 2^a.
There are infinitely many rational numbers, but there are infinitely more irrational numbers than rational numbers. There are more irrational numbers between 0 and 1 than there are rational numbers period.I was kind of guessing what you were trying to ask, so let me explain some background in case that wasn't quite it. Rational numbers are those that are representable as the ratio of two integers: 2/3, 355/113, 5 (=5/1). Irrational numbers are those that cannot be represented exactly by the ratio of two integers; some familiar irrational numbers are pi and the square root of 2. There are an infinite number of integers, and therefore an infinite number of rational numbers, but the two infinities are of the same order of magnitude (called a countable or listable infinity). The mathematical designation for the kind of infinity that the integers have is called aleph-null. There are also an infinite number of irrational numbers, but it's a "bigger" kind of infinity called C or the "power of the continuum." There's a relationship between aleph-null and a larger infinity called aleph-one. It's not known whether C and aleph-one are the same or not, and if they're not, we don't know which is bigger.
It could be:integersintegers greater than -27non-negative integers and -3rational numbersrational numbers greater than -3.2complex numbersThere are infinitely many possible answers.
Any real number is either rational or irrational. The rational ones are the ones that can be written in the form a/b where and b are integers and b does not equal 0. The irrational ones are all the other ones. If you expand your domain to include numbers other than the real numbers, like the imaginary numbers for example, there is no definition of "rational" or "irrational" for the non-real numbers. Zero is a rational number since it can be written as 0/1 and both 0 and 1 are integers.
0 and 1 (and -1) are the only integers which are not prime nor composite. All non-integral numbers are also non-prime and non-composite. This is because the property is not defined for such numbers.
Yes. The infinity of rational numbers has the same size as the natural numbers, said to be "countable". The infinity of real numbers (and therefore, also of irrational numbers) is a larger infinity, said to be "uncountable".
They are not rational, that is, they cannot be expressed as a ratio of two integers.Their decimal equivalent is infinitely long and non-recurring.Together with rational numbers, they form the set of real numbers,Rational numbers are countably infinite, irrational numbers are uncountably infinite.As a result, there are more irrational numbers between 0 and 1 than there are rational numbers - in total!
They are 6, 8, 10 and 12.
-- There's an infinite number of rational numbers. -- There's an infinite number of irrational numbers. -- There are more irrational numbers than rational numbers. -- The difference between the number of irrational numbers and the number of rational numbers is infinite.
The property of prime or composite is only defined for positive integers larger than 2.
There are more irrational numbers between any two rational numbers than there are rational numbers in total.
No. A rational number is ANY number that can be represented as one integer over a second integer (which cannot be zero). There is no requirement that the top integer is less than the bottom integer (an improper fraction is still a rational number - all integers are rational numbers as they can be represented as an improper fraction with a 1 as the denominator). Only if both rational numbers are less than 1 will the result of multiplying them together be less than both of them. If one rational number is greater than 1 and the other less than 1, then the result of multiplying them together is greater than the number less than 1 and less than the number greater than 1. If both rational numbers are greater than 1, then the result of multiplying them together is greater than both of them.
"Prime" or "composite" are terms that are defined for certain numbers, i.e., integers larger than 1. Thus, the concepts of "prime" or "composite" don't apply:* To complex numbers * To non-integers * To integers less than 2.
Irrational numbers are have decimal points that never stop. Like pi. They cannot be expressed as a fraction of two integers.Rational numbers, no matter how ridiculous and long, can be expressed in a fraction of two integers. If they are decimals, they will stop at some point. like 0.47They can't be both. That's a paradox. Unless your math teacher says so because he's probably smarter than me. :).................For rational numbers, think of the word ratio. A rational number can be expressed as the ratio (or fraction) of 2 integers. Irrational means not rational.
Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.Infinitely many. In fact, there are more irrational numbers between them than there are rational numbers.