Question

given z₁ and z₂, find the distance between them.
z₁ = 2 + i and z₂ = -4 - 3i
|z₁ - z₂| = ?√

Explanation:

Step1: Subtract \( z_2 \) from \( z_1 \)

To find \( z_1 - z_2 \), we substitute the given complex numbers:
\( z_1 - z_2=(2 + i)-(-4 - 3i) \)
Simplify the expression:
\( z_1 - z_2=2 + i + 4+ 3i \)
Combine like terms:
\( z_1 - z_2=(2 + 4)+(i + 3i) \)
\( z_1 - z_2 = 6 + 4i \)

Step2: Find the modulus of \( z_1 - z_2 \)

The modulus of a complex number \( a+bi \) is given by \( \sqrt{a^{2}+b^{2}} \). For \( z_1 - z_2 = 6 + 4i \), we have \( a = 6 \) and \( b = 4 \). But we can also factor out the greatest common factor of 6 and 4 first. The GCF of 6 and 4 is 2. So we can write \( 6 + 4i=2(3 + 2i) \). Then the modulus \( |z_1 - z_2|=|2(3 + 2i)| \). Using the property of modulus \( |k\cdot z| = |k|\cdot|z| \) where \( k \) is a real number and \( z \) is a complex number, we have \( |2(3 + 2i)|=|2|\cdot|3 + 2i| \). Since \( |2| = 2 \) and \( |3+2i|=\sqrt{3^{2}+2^{2}}=\sqrt{9 + 4}=\sqrt{13} \), but wait, actually we can directly compute the modulus of \( 6 + 4i \):
\( |z_1 - z_2|=\sqrt{6^{2}+4^{2}}=\sqrt{36 + 16}=\sqrt{52} \)
Now, we simplify \( \sqrt{52} \). We factor 52: \( 52=4\times13 \), so \( \sqrt{52}=\sqrt{4\times13}=2\sqrt{13} \)

Answer:

\( 2\sqrt{13} \) (So the first box is 2 and the second box is 13)