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Let z = a + ib it is in a complicated number. Then, the modulus of a facility number z, denoted by |z|, is identified to be the non-negative real number. $\sqrta^2 + b^2$modulus that a complicated number z = |z| = $\sqrtRe(z)^2 + Im(z)^2$where Real part of facility number = Re(z) = a andImaginary part of complicated number =Im(z) =b|z| = $\sqrta^2 + b^2$ .Example :(i) z = 5 + 6i therefore |z| = $\sqrt5^2 + 6^2$ = $\sqrt25 + 36$= $\sqrt61$(ii) z = 8 + 5i so |z| = $\sqrt8^2 + 5^2$ = $\sqrt64 + 25$= $\sqrt89$(iii) z = 3 - i so |z| = $\sqrt3^2 + (-1)^2$ = $\sqrt9 + 1$= $\sqrt10$(iv) z = 1 + $\sqrt5$i for this reason |z| = $\sqrt1^2 +\sqrt5^2$ = $\sqrt64 + 25$= $\sqrt89$(v) -6 + 2i so |z| = $\sqrt(-6)^2 + 2^2$ = $\sqrt36 + 4$= $\sqrt40$(vi) -8 + 6i therefore |z| = $\sqrt(-8)^2 + 6^2$ = $\sqrt64 + 36$= $\sqrt100$ = 10(vii) 12 - 5i for this reason |z| = $\sqrt12^2 + (-5)^2$ = $\sqrt144 + 25$= $\sqrt169$ =13

## Properties that Modulus of a complicated Number

Let z be any complicated number, then(I) |-z| = |z |Example : Let z = 7 + 8i-z = -( 7 + 8i)-z = -7 -8imodulus the (-z) =|-z| =$\sqrt(-7)^2 + (-8)^2$=$\sqrt49 + 64$ =$\sqrt113$modulus that (z) = |z|=$\sqrt7^2 + 8^2$=$\sqrt49 + 64$ =$\sqrt113$So indigenous the over we have the right to say that |-z| = |z |(II) |z| = 0 if, z = 0Proof : If z = a+ib ⇒ |z| = $\sqrta^2 + b^2$|z| = 0 ⇒ $\sqrta^2 + b^2$ = 0 ⇒ $a^2 + b^2$ = 0So, $a^2$ = 0 and also $b^2$ = 0 ⇒ a = 0 and also b = 0i.e. Z = 0 + i0 = 0So |z| = 0 if, z = 0(III) The pure of a product that two complicated numbers z1 and z2 is same to the product that the absolute worths of the numbers. I.e$\left |z1.z2 \right |$= $\left | z1 \right . |$ $\left | z2 \right |$(IV) The absolute of a quotient the two complicated numbers z1 and z2 (≠ 0) is equal to the quotient that the absolute values of the dividend and the divisor.$\left | \fracz1z2 \right |$= $\frac \left$(V) The pure of the amount of two conjugate complex numbers z1 and z2 deserve to never exceed the amount of their absolute values, i.e.$\left | z1+z2 \right |$ $\leq$ $\left | z1 \right |$ +$\left |z2 \right |$This inequality is referred to as triangle inequality.11th grade mathFrom modulus of a facility number to Home

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