simplify (1+3i)/(3+i) — step-by-step solution

1) Let's simplify:

\frac{1 + 3 i}{3 + i}

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

\frac{1 + 3 i}{3 + i}

\frac{(1 + 3 i) \times (3 - i)}{3^{2} + 1}

\frac{(1 + 3 i) \times (3 - i)}{3^{2} + 1}

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

3^{2}

3 \times 3

9

\frac{(1 + 3 i) \times (3 - i)}{9 + 1}

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

9 + 1

10

\frac{(1 + 3 i) \times (3 - i)}{10}

ⓘ Expand by multiplying each term inside the parentheses:

(1 + 3 i) \times (3 - i)

1 \times 3

+ 1 \times (- i)

+ 3 \times i \times 3

+ 3 \times i \times (- i)

\frac{1 \times 3 + 1 \times (- i) + 3 \times i \times 3 + 3 \times i \times (- i)}{10}

ⓘ Multiplications are done before additions and subtractions:

3 \times 3

9

■The minus sign is placed in front of the product:

3 \times (- i)

-3 i

\frac{3 - i + 9 i - 3 \times i \times i}{10}

ⓘ Like terms are grouped by adding their coefficients:

- i + 9 i

8 i

■Multiplying powers with the same base adds the exponents:

i \times i

i^{2}

\frac{3 + 8 i - 3 i^{2}}{10}

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

i^{2}

-1

■Multiplying by -1 gives the opposite:

- 3 \times (-1)

3

\frac{3 + 8 i + 3}{10}

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

3 + 3

6

\frac{6 + 8 i}{10}

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

\frac{6 + 8 i}{10}

\frac{3 + 4 i}{5}

\frac{3 + 4 i}{5}

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

\frac{3 + 4 i}{5}

\frac{3}{5} + \frac{4}{5} \times i

▶Simplified form:

\frac{3}{5} + \frac{4}{5} \times i

2) The numeric value of:

\frac{3}{5} + \frac{4}{5} \times i

equals:

▷

0.6 + 0.8 i