# Complex numbers such that modulus (absolute value) less than or equal to 1.

• MHB
• Taleb
In summary, for $u = a+bi$ and $v = c+di$, if $|u|$ and $|v|$ are both less than 1, then $a^2+b^2<1$ and $c^2+d^2<1$. When $u+v$ is calculated, it can be shown that $|u+v| < \sqrt{2+2(ac+bd)}$. It can also be proven by induction that $|u-v| \leq \sqrt{2-2(ac+bd)}$.
Taleb

Write u= a+ bi and v= c+ di. If modulus u and v are both less than 1 the $\sqrt{a^2+ b^2}< 1$ and $\sqrt{c^2+d^2}< 1$ so $a^2+ b^2< 1$ and $c^2+ d^2< 1$.

u+ v= (a+ c)+(b+ d)i. $|u+v|= \sqrt{(a+ c)^2+ (b+ d)^2}=$$\sqrt{a^2+ 2ac+ c^2+ b^2+ 2bd+ d^2}=$$ \sqrt{(a^2+ b^2)+ (c^2+ d^2)+ (2ac+2bd)}< \sqrt{1+ 1+ 2(ac+ bd)}< \sqrt{2+ 2(ac+ bs)}$

Can you prove that $ac+ bd$ is less than 1/2?

Following up on Country Boy's calculation, notice that if $v$ is replaced by $-v$ then $b$ becomes $-b$ and $d$ becomes $-d$. Therefore $$|u+v| \leqslant \sqrt{2+2(ac+bd)}, \qquad |u-v| \leqslant \sqrt{2-2(ac+bd)}.$$ It follows that if $ac+bd>0$ then $|u-v| \leqslant\sqrt2$, and if $ac+bd<0$ then $|u+v|\leqslant\sqrt2$. That proves 1). (In fact it proves a stronger result, with $\sqrt2$ instead of $\sqrt3$.)

Problem 2) seems to be a lot harder. I found a sketch here of how to prove it by induction (again with $\sqrt2$ rather than $\sqrt3$).

## 1. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are represented in the form a + bi, where a is the real part and bi is the imaginary part with i being the square root of -1.

## 2. What does it mean for a complex number to have a modulus less than or equal to 1?

The modulus of a complex number is its distance from the origin on the complex plane. When the modulus is less than or equal to 1, it means that the number is within a circle with a radius of 1 centered at the origin.

## 3. How are complex numbers with modulus less than or equal to 1 represented on a complex plane?

Complex numbers with modulus less than or equal to 1 are represented within a unit circle on the complex plane. The center of the circle is at the origin, and the numbers are located at various points along the circumference of the circle.

## 4. What is the significance of complex numbers with modulus less than or equal to 1 in mathematics?

Complex numbers with modulus less than or equal to 1 have many applications in mathematics, including in geometry, physics, and engineering. They are also used in solving equations and in signal processing.

## 5. How are operations performed on complex numbers with modulus less than or equal to 1?

Operations on complex numbers with modulus less than or equal to 1 are performed in the same way as operations on regular complex numbers. The only difference is that the result must also have a modulus less than or equal to 1.

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