# Contradiction in the definition and properties of the abs

• Jhenrique
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.
Jhenrique
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?

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?

$$|a + b| \neq |a| + |b|$$
E.g. a = 2, b = -1

1 person
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?

## 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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