Basic Properties of Numbers #1

Brief Explanation: Today we start with a different kind of post, one solely about Mathematics. Along with the more complicated exercises, I’ll be posting some simpler ones that I consider important as well.

Please be sure to check out this post in the blog Algebraic Can of Worms (A-CoW). The post presents a more intuitive approach to this exercise, relying on a great metaphor (frog leaps) to explain the relation between distance and absolute value!

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Question: Express the following with at least one less pair of absolute value signs.

Enunciado.png

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Comprehending: This is a very simple case but I want to state the 2 fundamental ideas to solve this exercise. First, the notion of absolute value:

Valor Absoluto.png

This means that the absolute value is always positive. If the expression inside is positive, the absolute value will have the same value as the expression. If the expression inside is negative, the absolute value will be that number multiplied by -1, making it positive. This is all that the absolute value means.

If we represent graphically the absolute value function we’ll get the following result:

Função Módulo.png

As you can see, the function is always positive (except on x = 0, obviously).

The other idea we need is the following inequality:

Desigualdade moduo.png

This is a basic idea that you must always remember. It will be very useful in the future in other things, such as epsilon-delta proofs. I’m not going to prove it here but it is quite simple: just take every case possible and think about the consequences (a>0 and b>0, a>0 and b<0, etc…).

Now we have all the ingredients to simplify the original expression.

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Solving:

First of all, let:

Captura de ecrã 2017-07-7, às 22.50.19.png

This is just for simplification. It will be easier and we will reduce the amount of letters in the expression. If we rewrite the original expression:

4.png

Now let’s analyze the second part of the expression. From the inequality we studied previously we know that:

3.png

This is just the inequality we analyzed in the Comprehending Section with different letters. If we subtract |d+c| from both sides:

4.png

This is all we need to know! Let’s return to the original notation:

5.png

Now let’s think for a bit: compare this expression with the original one. The original one has an extra set of absolute value bars. But remember: if the inside of the expression is bigger than zero, it is the same with or without these bars! 

In conclusion:

Conclusão.png

We were able to simplify and remove one set of absolute value bars.

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About the Problem: This is a very basic problem regarding the Basic Properties of Numbers but it makes you work with important concepts. I’ll be posting more problems like this with increasing difficulty. Please tell me what you think!

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3 thoughts on “Basic Properties of Numbers #1

  1. Pingback: The Basic Properties of Frogs #3.145 – Algebraic Can of Worms (A-CoW)

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