simplify (4+5i)/(4+6i) — step-by-step solution

1) Let's simplify:

\frac{4 + 5 i}{4 + 6 i}

ⓘ To make the denominator real, multiply the numerator and denominator by its conjugate, since (a+bi)(a-bi) = a²+b²:

\frac{4 + 5 i}{4 + 6 i}

\frac{(4 + 5 i) \times (4 - 6 i)}{4^{2} + 6^{2}}

\frac{(4 + 5 i) \times (4 - 6 i)}{4^{2} + 6^{2}}

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

4^{2}

4 \times 4

16

6^{2}

6 \times 6

36

\frac{(4 + 5 i) \times (4 - 6 i)}{16 + 36}

ⓘ The numerator and the denominator are computed before the division:

16 + 36

52

\frac{(4 + 5 i) \times (4 - 6 i)}{52}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{(4 + 5 i) \times (4 - 6 i)}{52}

\frac{(4 + 5 i) \times (2 - 3 i)}{26}

\frac{(4 + 5 i) \times (2 - 3 i)}{26}

ⓘ Expand by multiplying each term inside the parentheses:

(4 + 5 i) \times (2 - 3 i)

4 \times 2

+ 4 \times (-3) i

+ 5 i \times 2

+ 5 \times i \times (-3) \times i

\frac{4 \times 2 + 4 \times (-3) i + 5 i \times 2 + 5 \times i \times (-3) \times i}{26}

ⓘ Multiplications are done before additions and subtractions:

4 \times 2

8

5 \times 2

10

■Multiplications are done before additions and subtractions. A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

4 \times (-3)

-12

5 \times (-3)

-15

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{8 - 12 i + 10 i - 15 i^{2}}{26}

ⓘ Like terms are grouped by adding their coefficients:

- 12 i + 10 i

-2 i

■Multiplying by -1 gives the opposite:

- 15 \times (-1)

15

■i^2 = -1, so the powers of i reduce to 1, i, -1 or -i:

i^{2}

-1

\frac{8 - 2 i + 15}{26}

ⓘ The numerator and the denominator are computed before the division:

8 + 15

23

\frac{23 - 2 i}{26}

ⓘ A complex number is written in algebraic form a + b*i, a being its real part and b its imaginary part:

\frac{23 - 2 i}{26}

\frac{23}{26} - \frac{1}{13} \times i

▶Simplified form:

\frac{23}{26} - \frac{1}{13} \times i

2) The numeric value of:

\frac{23}{26} - \frac{1}{13} \times i

equals:

▷

\approx 0.8846 - \approx 7.6923 \times 10^{- 2} \times i