Complex numbers: arithmetic Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Complex numbers: arithmetic questions. See exactly how to solve problems on complex-numbers, addition-subtraction, multiplication, powers-of-i.

complex-numbersaddition-subtractionmultiplicationpowers-of-iconjugatesdivision
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Evaluate (3+2i)+(1+4i)\left(3+2i\right)+\left(1+4i\right).

Worked solution

  1. Write down the expression

    (3+2i)+(1+4i)\left(3+2i\right)+\left(1+4i\right)

    Both numbers are already in the form a+bia+bi.

  2. Group the real parts and the imaginary parts

    (3+1)+(2+4)i\left(3+1\right)+\left(2+4\right)i

    Real parts combine with real parts, imaginary with imaginary.

  3. State the answer in the form a+bia+bi

    (3+2i)+(1+4i)=4+6i\left(3+2i\right)+\left(1+4i\right)=4+6i

    This is the value of the expression.

Answer
4+6i4+6i
Question 2
2 markseasy
Evaluate (5i)+(2+3i)\left(5-i\right)+\left(2+3i\right).

Worked solution

  1. Write down the expression

    (5i)+(2+3i)\left(5-i\right)+\left(2+3i\right)

    Both numbers are already in the form a+bia+bi.

  2. Group the real parts and the imaginary parts

    (5+2)+(1+3)i\left(5+2\right)+\left(-1+3\right)i

    Real parts combine with real parts, imaginary with imaginary.

  3. Work out each bracket

    7+2i7+2i

    This is the answer in the form a+bia+bi.

  4. State the answer in the form a+bia+bi

    (5i)+(2+3i)=7+2i\left(5-i\right)+\left(2+3i\right)=7+2i

    This is the value of the expression.

Answer
7+2i7+2i
Question 3
2 markseasy
Evaluate (7+3i)(2+i)\left(7+3i\right)-\left(2+i\right).

Worked solution

  1. Write down the expression

    (7+3i)(2+i)\left(7+3i\right)-\left(2+i\right)

    Both numbers are already in the form a+bia+bi.

  2. Group the real parts and the imaginary parts

    (72)+(31)i\left(7-2\right)+\left(3-1\right)i

    Real parts combine with real parts, imaginary with imaginary.

  3. Work out each bracket

    5+2i5+2i

    This is the answer in the form a+bia+bi.

  4. State the answer in the form a+bia+bi

    (7+3i)(2+i)=5+2i\left(7+3i\right)-\left(2+i\right)=5+2i

    This is the value of the expression.

Answer
5+2i5+2i
Question 4
2 markseasy
Evaluate (45i)(62i)\left(4-5i\right)-\left(6-2i\right).

Worked solution

  1. Write down the expression

    (45i)(62i)\left(4-5i\right)-\left(6-2i\right)

    Both numbers are already in the form a+bia+bi.

  2. Group the real parts and the imaginary parts

    (46)+(5+2)i\left(4-6\right)+\left(-5+2\right)i

    Real parts combine with real parts, imaginary with imaginary.

  3. State the answer in the form a+bia+bi

    (45i)(62i)=23i\left(4-5i\right)-\left(6-2i\right)=-2-3i

    This is the value of the expression.

Answer
23i-2-3i
Question 5
2 markseasy
Evaluate (2+7i)+(53i)\left(-2+7i\right)+\left(5-3i\right).

Worked solution

  1. Write down the expression

    (2+7i)+(53i)\left(-2+7i\right)+\left(5-3i\right)

    Both numbers are already in the form a+bia+bi.

  2. Group the real parts and the imaginary parts

    (2+5)+(73)i\left(-2+5\right)+\left(7-3\right)i

    Real parts combine with real parts, imaginary with imaginary.

  3. Work out each bracket

    3+4i3+4i

    This is the answer in the form a+bia+bi.

  4. State the answer in the form a+bia+bi

    (2+7i)+(53i)=3+4i\left(-2+7i\right)+\left(5-3i\right)=3+4i

    This is the value of the expression.

Answer
3+4i3+4i

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