What is a number that is not rational?

An example of such a number is  ("Square root of 2").  It is not possible to name any whole number, any fraction or any decimal whose square is 2. 7/5 is close because 7/5 multiplied by 7/5 equals to 49/25, which is almost 2.

To prove that there is no rational number whose square is 2, suppose there was. Then we could express it as a fraction m/n in lowest terms.

That is, suppose m/n multiplied by m/n equals to m multiplied by m divided by n multiplied by n equals 2. But that is impossible. Since m/n is in lowest terms, then m and n have no common divisors except 1.  Therefore, m· m and n· n also have no common divisors – they are relatively prime – and it will be impossible to divide n· n into m· m and get 2.

There is no rational number – no number of arithmetic – whose square is 2.  Therefore we call   an irrational number.

Task 13

Something about mathematical symbols:

= When two values are equal, we use the "equals" sign, for example:  2+2 = 4

≠ When two values are definitely not equal, we use the "not equal to" sign, for example: 2+2 ≠ 9

< But when one value is smaller than another, we can use a "less than" sign, for example: 3 < 5

> And when one value is bigger than another, we can use a "greater than" sign, for example: 9 > 6

Solve the problems:

1. What is x + 4, when you know that x is greater than 2?

2. What is x – 5, when you know that x is less than 7?

3. What is x + 6, when you know that x is less than or equal to 3?

4. What is x – 3, when you know that x is greater than or equal to 9?

5. What is x + 6, when you know that x is greater than or equal to 3?

6. What is x – 3, when you know that x is less than or equal to 9?

7. What is x – 5, when you know that x is greater than 7?

Task 15

Read the answer which a Math sophomore gave to the question of a fresher: "Why do we need rationals and irrationals?". Do you agree? If you do, isn’t there anything to add? If you don’t, write your own answer.

I think irrational numbers are for specifying an amount of something accurately(or far more specific if not accurate) up to infinitely small scale and can never be reached by getting the ratio of any 2 whole numbers(integers). Examples: amount of liquid, exact volume of a container, exact length of a rope around a drum, density of moisture in the air. (Though, when computing these quantity, we get rational numbers. But that can't be easily known.)

Rational numbers occur when counting some whole/countable objects like apple, orange, houses, trees, ballpens, etc, are involve, which getting the count of an object as a whole(ignoring their sizes) are more important than getting their volume, or the space they occupy nor their weight.

Example: "How many coins do you have right now?" ask for a rational number while "How much profit you earned in your bank last year?" ask for a quantity that has a whole number count and a an amount less than a whole number to specify exactness.

Non-integer rational numbers still involves whole numbers. Example: I have apples double your number of apples, or your apples are just 50% of my apples. We know that 50% comes from 50/100 or 1/2. If we know these, 50% or 0.5 is obviously a rational number. They are accurate if expressed as fractions compared to floating point notation which involves rounding off when writing them to save space.

I think, rational is more on finite measurement while irrational is more on infinite measurement(infinitely small approximations). I'm just answering based on my opinion and have no complete info about these two set of numbers. But for informal explanation that is my answer. I hope it helps.

Enjoy yourself!

Natural Numbers

– What do you know about natural numbers?

– Natural numbers are better for your health!

Obedient Kids

Teacher: Now class, whatever I ask, I want you all to answer at once. How much is six plus 4?

Class: At once!

Sisters

Question: Two girls were born to the same mother, at the same time, on the same day, in the same month and in the same year and yet somehow they’re not twins. Why not? 

Answer: Because there was a third girl, which makes them triplets!