# If $z$ and $w$ are complex numbers can we use the proof in $\mathbb{R}$ to demonstrate that $|z w|=|z||w|$?

If yes could you explain why? Sorry if the question is trivial but I'm new to complex numbers and I see lots of examples where properties of real numbers are used in complex without to prove it . This really surprise me since complex numbers are a superset of real numbers. For example I saw in my book $z*1=z$ in $\mathbb{C}$ I had to make the complex multiplication to be sure that it was true, because this 1 is in $\mathbb{C}$ (1,0)

• What do you mean by "the proof in $\Bbb R$"? Also, what is your definition of $|z|$? I know of at least two (equivalent) definitions, and I would like to know which one you use. – Wojowu Nov 11 '15 at 11:44
• I thought in $\mathbb{C}$ the only that exists is this $|z|=\sqrt{z \overline{z} }$ – GniruT Nov 11 '15 at 11:49
• You could also define it as $|a+bi|=\sqrt{a^2+b^2}$. – Wojowu Nov 11 '15 at 11:50
• HINT assume the numbers $z=x+iy , w=a+ib$ and then separately find out LHS and RHS. Apply basic algebra . Hope you can take it from there and also hope you know how to find out the mod. – Archis Welankar Nov 11 '15 at 11:50
• Related recent post: How to prove $|z|\cdot |w|=|z \cdot w|$ form complex numbers? – Martin Sleziak Nov 6 '17 at 8:56

First note that, for $z=a+ib$ we have $z\overline{z}=\sqrt{a^2+b^2}=|z|^2$ and also $\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}.$
Therefore, $|zw|^2=(zw)\overline{zw}=zw\overline{z}\,\overline{w}=z\overline{z}w\overline{w}=|z|^2|w|^2.$ This implies $|zw|=|z||w|.$
Let $z=a+bi$ and $w=c+di$ where $(a,b,c,d)\in\mathbb{R}^{4}$. The module $|z|$ is defined by $|z|=\sqrt{a^{2}+b^{2}}$ and $|w|=\sqrt{c^{2}+d^{2}}$, thus \begin{align} |z||w| &=\sqrt{a^{2}+b^{2}}\sqrt{c^{2}+d^{2}}\\ &=\sqrt{a^{2}c^{2}+a^{2}d^{2}+b^{2}c^{2}+b^{2}d^{2}} \end{align}
Now, consider $zw=(a+bi)(c+di)=(ac-bd)+(ad+bc)i$. We have
\begin{align} |zw|&=\sqrt{(ac-bd)^{2}+(ad+bc)^{2}}\\ &=\sqrt{a^{2}c^{2}+b^{2}d^{2}-2abcd+a^{2}d^{2}+b^{2}c^{2}+2abcd}\\ &=\sqrt{a^{2}c^{2}+b^{2}d^{2}+a^{2}d^{2}+b^{2}c^{2}}\\ &=|z||w| \end{align}