As per some sources fractions have been defined as a quotient of two numbers whereas some sources restrict the numerator to be a whole number and denominator to be a positive integer. What is the exact definition?
2026-04-04 02:21:33.1775269293
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What is the exact mathematical definition of a fraction? Can all real numbers be expressed as fractions?
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A fraction (from Latin: fractus, "broken") represents a part of a whole - Wikipedia
More simply, a fraction is a notation. Let's say there are 5 apples. You have eaten 3. A way of expressing this statement is that $\frac{3}{5}$ of the 5 apples have been consumed.
A form of a fraction which can be used for operations is known as a rational number. By definition, a rational number is a number which can be represented by $\frac{p}{q}$ where $q≠0$ and $p,q∈Z$. The fraction $\frac{3}{5}$ can be represented as a rational number. But all fractions cannot be represented as rational numbers
A good example includes this. You have a square of area 4 $m^2$. You cut out a circle of radius 1 m it. The area of the circle will be $π m^2$. So the area of your circle is $\frac{π}{4}$ times the total area of the square.
But $\frac{π}{4}$ is not a rational number, as π is an irrational number and cannot be expressed as $\frac{p}{q}.$
Like π, other irrational numbers like $\sqrt{2},\sqrt{3}$ cannot be expressed as rational numbers.
Fractions can never be negative as well.
PS:- A Fraction is a notation. You cannot express a real value as a fraction, but as a rational number or irrational number.
A fraction is generally anything that can be written in the form $\frac {something}{otherthing}$ where the value is a ration between the two things, or another way of thinking about it, is one thing divided by another. For instance $\frac 8{16}$ is the ratio of $8$ to $16$ of the value of $8\div 16$ and has the value of one-half, or if you want to use decimals (as your text obviously wants to do), the value of $0.5$.
Another example is if you take a circles circumference $C$ and it's diameter $d$ the the ratio of $C$ to $d$ or $\frac Cd$ is will always have the ratio we call $\pi$ with $\pi =\frac Cd$. $\frac Cd$ is a fraction. If we wanted to divide $\pi$ to a number one-fourth it's size we could write $\frac \pi 4$.
As you can see a fraction can be between any two real numbers. We can even write a fraction as $\frac 70$ even though that is no such possible value. It is an invalid and wrong fraction and can not be any possible value but it is still a fraction. Just a wrong and useless one.
And, because $w\div 1 = w$, we can always have $w = \frac w1$ and so, yes, every real number can be a fractions.
Fractions are not a very useful concept.
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BUT!!!
There is an entire class of real numbers called the rational numbers. A rational number is any real value that can be written as a fraction BETWEEN WHOLE NUMBERS. Or to be more precise whose numerator is an integer, and whose denominator is a positive (not zero) integer.
$\frac 8{17} = 0.470588235294117647058823529470588235294117647058823529470588235294117647058823529...$ is are rational number because $8$ and $17$ are integers. But $\frac \pi 4$ is not because $\pi$ is not an integer. And there is no way we can change the values and make it a fraction between whole numbers.
Rational (and irrational) numbers are a very important concept.