4.4 Operations on Complex Numbers

Front

What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as i = √(-1). It is used to represent the square root of negative numbers.

Front

How do you add complex numbers?

Back

To add complex numbers, combine their real parts and their imaginary parts separately. For example, (-9 + 5i) + (3 - 2i) = (-9 + 3) + (5i - 2i) = -6 + 3i.

Front

What is the formula for multiplying two complex numbers?

Back

To multiply two complex numbers (a + bi)(c + di), use the distributive property: ac + adi + bci + bdi², and remember that i² = -1.

Front

What is the result of (11 + 5i)(11 - 5i)?

Back

The result is 146, calculated as (11 + 5i)(11 - 5i) = 11² - (5i)² = 121 - (-25) = 121 + 25 = 146.

Front

How do you multiply a complex number by a real number?

Back

To multiply a complex number by a real number, multiply the real part and the imaginary part by that real number. For example, 3i(4 - i) = 3i * 4 - 3i * i = 12i + 3 = 3 + 12i.

Front

What is the value of i^2?

Back

The value of i^2 is -1.

Front

What is the pattern of powers of i?

Back

The powers of i cycle every four: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then it repeats.

Front

What is the simplified form of i^13?

Back

The simplified form of i^13 is i, as 13 mod 4 = 1.

Front

How do you square a complex number?

Back

To square a complex number (a + bi)², use the formula: (a + bi)² = a² + 2abi + (bi)² = a² - b² + 2abi.