Monday, November 7, 2011

A more complete proof that the square root of 2 is irrational.

I don't know about you but I never quite liked the usual proof by contradiction that the square root of 2 is irrational. It seems incomplete in some way. I never felt convinced by it. Here's what I feel is the missing piece of the puzzle.

Assume that the square root of 2 is rational. So,
√2 = a/b
2 = (a/b)^2
2 = a^2 / b^2
2 b^2 = a^2

So far so good. The usual proof continues with the following statement:

Since a^2 is equal to a natural number multiplied by 2, a^2 is an even number. But for a^2 to be even, a must be even too (see proof in the appendix at the end). So that means that there is a natural number k where a = 2k.

Since a = 2k and 2 b^2 = a^2,
2 b^2 = a^2
2 b^2 = (2k)^2
2 b^2 = 4 k^2
b^2 = 2 k^2

Just like for a, b must also be an even number.

The proof usually ends right there, claiming that since a and b are both even numbers, then the fraction a/b is not simplified and irreducible, contradicting that a/b exists. But let's see where the proof takes us if we just keep on going.

If both a and b are even, then the fraction a/b can be simplified by dividing both a and b by 2, that is, if a = 2k and b = 2l, then we can say that √2 = k/l. But after doing this we can reapply the same reasoning on k and l and we'll discover that k and l are also both even numbers, and we can do it again and again ad infinitum.

So, which natural numbers can be divided by 2 infinitely? Only 1 number can do that, zero. But replacing zero for both a and b will not make their quotient a real number, or if you want to define 0/0, it will not result in a number whose square equals 2. So there is no fraction a/b which gives √2.

So there you have it, a proof that goes on till the end.

Now on to the proof that an even square can only come from an even number squared:

Let a^2 be an even number.

a can either be even or odd, that is there must exist an n where
a = 2n or a = 2n + 1
If a = 2n, a^2 = (2n)^2 = 4 n^2 = 2(2 n^2), which is an even number
If a = 2n + 1, a^2 = (2n + 1)^2 = 4 n^2 + 4n + 1 = 2(2 n^2 + 2n) + 1, which is an odd number

So an even number squared will give an even number and an odd number squares will give an odd number. Hence, a square even number can only come from an even number squared.

Wednesday, November 2, 2011

Wisdom hierarchy vs Bloom's taxonomy

So lately I've been reading about two subjects that I noticed are very related, the Wisdom hierarchy ( and Bloom's taxonomy ( First I need to explain each.

Wisdom hierarchy
This is a hierarchy of how wisdom is obtained and describes the relationship between data, information, knowledge, understanding and finally wisdom.

Data is symbols and signals which can be observed and analysed, but perhaps not be processed and organized.
An example of this is seeing the symbols "3", "×", "4", "=" and "12". Those symbols may not mean anything to you if you don't know arithmetic.

Information is data which is given meaning and use. It answers "what", "where", "who" and "when" questions, that is, simple shallow questions. It is when relationships are formed between the different data and context is given to the data. The data has meaning but perhaps it cannot be used.
So now "3×4=12" has a meaning. It means that if you multiply the numbers 3 and 4, the result is equal to 12. You may know what the symbols mean but you may not be in a form that is useful.

Knowledge is a mass of information which is organized in a way to be useful. It answers "how" questions, that is how can I use the information. The information may be useful but perhaps you don't understand why it is related and how to generate new information from it.
So now we have organized every multiplication of two numbers we learned into a multiplication table. If we want to know what a particular multiplication equals, we know how to do that, we simply look it up our multiplication table. You may know how to multiply numbers together but you may not know why when numbers are multiplied they give a particular number as a result.

Understanding is when you understand the knowledge, when you find a pattern to the organization and can use the pattern to generate new information. It answers "why" questions, that is, why is the information organized as it is in the knowledge. The knowledge may be understood but perhaps it cannot be judged and compared with other knowledge.
So now we understand that multiplication is repeated addition. Now we can add to our knowledge new information which is generated from our understanding rather than from the external world (such as having to ask someone). You may understand how to do multiplication but you may not be able to compare different methods to doing multiplication.

Wisdom is when you can pass judgement and make decisions to determine what is the best method to use. The question it could answer is "which" questions, that is, which is best.
We now can decide which method we should use to multiply two numbers, be it by looking up the multiplication table, by repeated addition or by long multiplication.

Bloom's taxonomy
This is a way of categorizing exam questions in a hierarchy such that as you go up the pyramid, the higher the level of thought required to answer the question.

Remembering type questions are those that only require the student to remember things, without expecting any understanding.
An example question would be "What does the symbol × represent?".

Understanding type questions are those that require the student to know what the things they know actually mean.
An example question would be "Explain what the expression 2×3=6 means in your own words.".

Applying type questions are those that require the student to be able to use what they know in a situation.
An example question would be "How many apples would you have if you had 2 baskets with 3 apples in each?".

Analyzing type questions are those that require the student to break down a problem into parts and see how they are related to each other.
An example question would be "What is the next number in the sequence 21, 42, 63, __".

Evaluating type questions are those that require the student to justify a decision.
An example question would be "Which multiplication method would you use to multiply 128 by 64 and why?".

Creating type questions are those that require the student to create something new to the student.
An example question would be "If all you have is the product of the sum and difference of two numbers and one of the numbers, how can you find the other number?".

It is clear that there is a relationship between the two hierarchies. We could say that:
Remembering type questions test the student having memorized data.
Understanding type questions test if the student has derived information from data.
Applying type questions test if the student has developed a useful knowledge from the information and if the knowledge can be readily used.
Analysis type questions test if the student has understood the basis of their knowledge and can derive new information from it.
Evaluating type questions test if the student has obtained any wisdom on the subject and hence can make sound judgement about it.

The last question type, creating, is not covered by the Wisdom hierarchy and perhaps it predicts yet another higher level form of cognition, perhaps called "creativity", which is when you use knowledge, understanding and wisdom together to derive new knowledge, understanding and wisdom, where knowledge provides the raw material to act on, understanding provides the ways to rearrange the knowledge and wisdom guides you into choosing a solution path which is most likely to give good results. Once this is done you will have learned from experience and would have added new knowledge, a deeper understanding of that knowledge together with new ways of using it and you would be able to make better judgement in the future.