Contradiction in the definition and properties of the abs

In summary, the conversation discusses the equation abs(a+b) = abs(a) + abs(b) and questions whether abs(z) = √[x²+y²] is a correct statement. It is concluded that this equation is incorrect and any following work is meaningless. It is also noted that the | key on the keyboard is usually used to write absolute values.
  • #1
Jhenrique
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If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?
 
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  • #2
Jhenrique said:
If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?

[tex]|a + b| \neq |a| + |b|[/tex]
E.g. a = 2, b = -1
 
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  • #3
Jhenrique said:
If the abs(a+b) = abs(a) + abs(b)
As PeroK notes, this is incorrect. Since your hypothesis is false, any following work is meaningless.

Also, the | key on the keyboard is usually used to write absolute values, as in |a + b|.
Jhenrique said:
, so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?
 

Related to Contradiction in the definition and properties of the abs

1. What is the contradiction in the definition of the abs?

The contradiction in the definition of the abs, or absolute value, is that it is defined as the distance of a number from zero on a number line, yet it always results in a positive value.

2. How can the abs be both a distance and always positive?

This contradiction can be better understood by thinking of the abs as a concept rather than a specific number. When we think of distance, we usually think of it as a positive value. However, in math, distance can be both positive and negative depending on the direction. The abs simply removes the negative sign and gives us the positive distance.

3. Is the abs a function or an operator?

This is a matter of interpretation and convention. Some mathematicians consider the abs to be a function, as it takes in a number and gives back a value. Others consider it to be an operator, as it operates on a number and changes it in some way. Both interpretations are valid.

4. Can the abs of a complex number be negative?

No, the abs of a complex number is always a positive real number. This is because the abs measures the distance from zero on a number line, and a complex number is a combination of a real and an imaginary number. The distance from zero is always positive.

5. What are the properties of the abs?

The properties of the abs include:

  • Always results in a positive value
  • Commutative: |a| = |b| if and only if a = b
  • Multiplicative: |ab| = |a| x |b|
  • Triangle inequality: |a + b| ≤ |a| + |b|
  • Can be extended to complex numbers

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