Question

I'm a freshman and have daily cry sessions before and after my Algebra I classes. What the heck is Rational and Irrational numbers??? And fractions???

Answer 1

a fraction is a representation of a proportion, like say you have a pizza and cut it in 6 pieces, each piece is 1/6 of the pizza, $\bf \cfrac{\stackrel{\textit{how many units}}{1}}{\stackrel{\textit{total available units}}{6}}$.

the total number of available slices in the pizza is 6, you cut it in 6 slices, and a single unit of it or just 1, is 1/6 or one sixth of the pizza.

in short, a rational number is one you can write as a fraction, and an irrational number is one that can't be written as a fraction, a good example of those are π , or e euler's constant or √2.

Answer 2

You might find both comfort and education by spending a little time in the kitchen cooking from recipes. You will learn that measuring spoons and cups are related in size by fractions. That is, you will find a 1 cup measure, a 3/4 cup measure, a 2/3 cup measure, a 1/2 cup measure, along with 1/3 and 1/4 cup measures. Likewise for teaspoons: the usual set includes 1/2, 1/4, and 1/8 teaspoon measures.

A fraction has parts that are named:

$\text{fraction}=\dfrac{\text{numerator}}{\text{denominator}}$

The teaspoon measures described above all have a numerator of 1. The denominator tells how many of that size are necessary to make up one whole teaspoon. That is, it takes 4 of 1/4 teaspoon to be the same amount as 1 teaspoon.

When the numerator is not 1, it tells you how many of the parts of size (1/denominator) you are talking about. That is, a 2/3 cup measure is the same volume as two 1/3 cup measures. In math terms, we might say

$\dfrac{2}{3}=2\times\dfrac{1}{3}$

This also gives us a clue as to how we do math with fractions.

You can compare the sizes of fractions fairly easily with measuring cups. You can put the entire contents of a 2/3 cup measure into a 3/4 cup measure, for example, but not the other way around. (3/4 is larger than 2/3). You can use a 1/2 cup measure together with a 1/4 cup measure to fill a 3/4 cup measure. That is 1/2 is 2 quarters, and 3/4 is 2/4 + 1/4. This, too, gives a clue as to how we do math with fractions.

_____

We don't often deal with cash money anymore, but that, too, can help you see and remember what's going on.

For example, it takes four (4) quarters to make 1 dollar. The coin has written on it that it is a quarter dollar. Another way to write "one quarter" is "1/4". The 4 in the denominator tells you this is 1 of 4 equal parts (of a dollar, in this case).

— — — — —

If you actually do this exercise, you will find that liquids are funny things to measure (they can be heaped up), and that dry things need to be carefully leveled without packing (or packed consistently).

— — — — —

A fraction is called improper if its numerator is larger than its denominator. For example, 3/2 is an improper fraction, as is 9/4. An improper fraction can be converted to the sum of an integer and a proper fraction (and vice versa). Such a sum, written without the + sign, is called a mixed number. Our last example is

$\dfrac{9}{4}=\dfrac{8}{4}+\dfrac{1}{4}=2+\dfrac{1}{4}=2\frac{1}{4}$

The fraction, or ratio, 9/4 ("nine fourths") also means "nine divided by 4", so will have the value that is the usual result of that division: 2.25. Here, we see that 2.25 is the same as the mixed number 2 1/4.

_____

A rational number and a fraction are the same thing: one integer divided by another integer (except the denominator cannot be zero). An irrational number is a number than cannot be expressed exactly as a rational number.

A rational number written in decimal form will be one that (a) terminates (has a finite number of digits), or (b) repeats (has a finite number of digits that repeat themselves endlessly). For example, 31/10 = 3.1, a decimal that has one digit after the decimal point. 22/7 = 3.142857142857... with the sequence 142857 repeating endlessly.

An irrational number is one that has no repeating sequence and does not end (has an infinite number of digits). The square root of 2 is one such number. Pi is another famous one. Some of these numbers have been calculated out to trillions of digits, but that's not the end of it—because there is no end.

__ __ __ __ __

Fractions are pretty useful in algebra, so it is helpful to become friends with them. One of the useful bit about fractions is that when the numerator and denominator are equal, the value is 1. We know that multiplying by 1 does not change the value of anything.

We already showed that multiplying a fraction by an integer just mutiplies the numerator. When you're multiplying a fraction by a fraction, the numerator multiplies the numerator, and the denominator multiplies the denominator. Here's an example:

$\dfrac{2}{7}\times\dfrac{3}{5}=\dfrac{2\times 3}{7\times 5}=\dfrac{6}{35}$

This is true for any kind of fractions, whether they are proper or improper. This means we can turn a fraction into 1 by multiplying by the same fraction with the numerator and denominator switched:

$\dfrac{2}{3}\times\dfrac{3}{2}=\dfrac{2\times 3}{3\times 2}=\dfrac{6}{6}=1$

This is handy when we want to solve an equation like

$\dfrac{1}{3}a=4\\\\ \dfrac{3}{1}\times\dfrac{1}{3}a=\dfrac{3}{1}\times 4\qquad\text{multiply both sides by the inverse of 1/3}\\\\ a=\dfrac{12}{1}=12$

There's a lot more to learn about fractions and how to change them from one form to another, but this can get you started. (Please don't cry in the cookie dough.)