# Need materials to learn about different types of fractions

I wish i had one book or pdf to learn about different types of fractions .

fresh_42
Mentor
What do you mean by type of fractions? Which are the kind of difficulties you have with fractions?

fresh_42
Mentor
I would suggest to forget about these categories like "proper" and "improper" or similar. I also suggest to get rid of notations like ##1\frac{1}{2}## for what is meant to be ##\frac{3}{2}##. It is a terrible bad habit which apparently can't be eliminated. The reason why I object it is the following:

Whenever a sign is missing, like in ##5\, x## or ##0.25\, m## or ##68\, kg## it is a multiplication sign: ##5 \cdot x\; , \;0.25 \cdot m \; , \; 68 \cdot kg##. But in this only example ##1\frac{1}{2}=\frac{3}{2}=1\,+\,\frac{1}{2}## it is an addition which is omitted. This is a completely unnecessary confusion and the reason I really don't like this notation. As a consequence, the classification "improper" is nonsense.

I find it far better to be aware of the fact, that a division like a quotient is actually a multiplication, the multiplication with an inverse element. Take any number say ##8##. Then the inverse element is ##\frac{1}{8}##. It simply means ##8 \cdot \frac{1}{8} = 1##. This means the inverse is the number we need in a multiplication with the number to obtain ##1##. With this in mind, you can always write a division as a multiplication with an inverse element, e.g. ##\frac{3}{2}=3 \cdot \frac{1}{2}##. A fraction, whether it is written by ##":"## or ##" \div "## is always a division, resp. a multiplication with the inverse to the number in the denominator.

In general you have to learn the following rules (if that fits you better):
$$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + b \cdot c}{b \cdot d}\,$$ $$\, \frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - b \cdot c}{b \cdot d}$$

$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \,$$ $$\, \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} : \frac{c}{d} = \frac{a \cdot d}{b \cdot c}$$
where ##a,b,c,d## are any numbers (denominators not zero!). To avoid larger numbers or to solve additions and subtractions, it is also helpful to remember
$$\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$$

can help you (it's free). If you don't give up you will certainly find sources on the internet that can help you learning it, just don't give up. I know that many serious and less serious offers are out there to make money out of student's tutorials, so watch out for hidden costs and traps. If something looks suspicious: LEAVE! There are still pages, which are for free like ours. If you have questions to some of your exercises, try our homework section https://www.physicsforums.com/forums/precalculus-mathematics-homework.155/ and ask!

• awholenumber
Thanks a lot fresh_42 ,
That is a very helpful post .

I was trying to follow this book after searching for a very long time , because i needed to improve a lot of things from basics

http://ncert.nic.in/ncerts/l/feep104.pdf

Other than that book , i have not come across any quality materials on fractions

Now time to practice from a lot of sources like Khan Academy

Mark44
Mentor
As a consequence, the classification "improper" is nonsense.
The way I remember it is that in an improper fraction, the numerator is greater than or equal to the denominator, so 3/2 would be an improper fraction. A number like 1 1/2 is called a mixed number. I agree that mixed numbers really have no place in mathematics.

Other than that book , i have not come across any quality materials on fractions
About all you need to know about fractions is covered in @fresh_42's post #4: i.e., how to add, subtract, multiply, and divide fractions. Don't overthink this.

fresh_42
Mentor
The way I remember it is that in an improper fraction, the numerator is greater than or equal to the denominator, so 3/2 would be an improper fraction.
Yes, that's also what is said in the OP's link. But what is it good for? I consider it ballast. I cannot imagine a case where it is necessary to name quotients ##\frac{4}{5}## and ##\frac{5}{4}## differently. It seems only to be necessary if one wants to write the improper ones as "mixed numbers". But in the end it's an opinion. Dropping a "+" sign in a framework which usually drops "##\cdot ##" is questionable at least, proper and improper ... whomever it fits. It wouldn't be my first choice when learning about fractions.

Mark44
Mentor
Yes, that's also what is said in the OP's link. But what is it good for? I consider it ballast.
I agree completely.
fresh_42 said:
I cannot imagine a case where it is necessary to name quotients ##\frac{4}{5}## and ##\frac{5}{4}## differently. It seems only to be necessary if one wants to write the improper ones as "mixed numbers". But in the end it's an opinion. Dropping a "+" sign in a framework which usually drops "##\cdot ##" is questionable at least, proper and improper ... whomever it fits. It wouldn't be my first choice when learning about fractions.
The only place where this concept comes into play again is in Partial Fraction Decomposition, which could be used in this integral: ##\int \frac{x^2}{x^2 - 1}##. Here the integrand is an improper rational function. Writing the integrand as ##\frac A {x^2 - 1} + \frac B {x^2 - 1}## won't work, but using polynomial division or a clever trick, one could write the integral as ##\int [1 + \frac 1 {x^2 - 1}]dx##, and then break this into two integrals and use partial fractions or a trig substitution on the last part.
As far as ordinary numbers go and in the context of the OP's question, I don't see any good reason in writing mixed numbers such as 2 1/3 in any mathematical setting.

• awholenumber and fresh_42
Mark44
Mentor
I wish i had one book or pdf to learn about different types of fractions .
If there any such books, they must be very thin; there's not very much to say about fractions. Any book on arithmetic should have a section on fractions.

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