# B Need materials to learn about different types of fractions

1. Jun 11, 2017

### awholenumber

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

2. Jun 11, 2017

### Staff: Mentor

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

3. Jun 11, 2017

### awholenumber

4. Jun 11, 2017

### Staff: 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!

5. Jun 11, 2017

### 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

6. Jun 11, 2017

### Staff: 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. A number like 1 1/2 is called a mixed number. I agree that mixed numbers really have no place in mathematics.

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.

7. Jun 11, 2017

### Staff: Mentor

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.

8. Jun 11, 2017

### Staff: Mentor

I agree completely.
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.

9. Jun 11, 2017

### Staff: Mentor

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.