adding subtracting multiplying and dividing complex numbers

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What is a complex number?

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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.

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CCSS.HSN.CN.A.1

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How do you add two complex numbers?

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To add two complex numbers, you add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

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CCSS.HSN.CN.A.2

Front

How do you subtract two complex numbers?

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To subtract two complex numbers, you subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

Tags

CCSS.HSN.CN.A.2

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What is the formula for multiplying two complex numbers?

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To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i.

Tags

CCSS.HSN.CN.A.2

Front

How do you divide two complex numbers?

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To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide (a + bi) by (c + di), multiply by (c - di) to get: (a + bi)(c - di) / ((c + di)(c - di)).

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CCSS.HSN.CN.A.3

Front

What is the conjugate of a complex number?

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The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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CCSS.HSN.CN.A.3

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What is the modulus of a complex number?

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The modulus of a complex number a + bi is given by the formula |a + bi| = √(a² + b²). It represents the distance of the complex number from the origin in the complex plane.

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How do you find the square of a complex number?

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

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What is the result of (2 + 3i)²?

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The result of (2 + 3i)² is -5 + 12i.

Tags

CCSS.HSN.CN.C.8

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How do you simplify the expression 3i - (4 + 7i)?

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To simplify 3i - (4 + 7i), distribute the negative sign: 3i - 4 - 7i = -4 - 4i.

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CCSS.HSN.CN.A.2