Complex Numbers (Optional)

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities.

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions.

ND

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions. Specifically the denominator D instructs us to divide a whole item into D equal parts then we are to take N of these parts.

ND

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions. Specifically the denominator D instructs us to divide a whole item into D equal parts then we are to take N of these parts.

ND

Fractions provide answers for all equations of the form Ax + B = 0 (where A, B are integers)

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions. Specifically the denominator D instructs us to divide a whole item into D equal parts then we are to take N of these parts.

ND

Fractions provide answers for all equations of the form Ax + B = 0 (where A, B are integers)

However as we’ve seen that there is no fractional answer for x2 = 2

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions. Specifically the denominator D instructs us to divide a whole item into D equal parts then we are to take N of these parts.

ND

Fractions provide answers for all equations of the form Ax + B = 0 (where A, B are integers)

However as we’ve seen that there is no fractional answer for x2 = 2 So we invented the symbol 2 ≈ 1.414.. to track this quantity and we call these numbers irrationals.

Complex NumbersLet’s briefly retrace the development of numbers- the system that we invented to track quantities. In the beginning, integers such as 1, 2, 3… were invented to count whole items such as chickens and cows.Next we use two integers N and D, recorded as N/D or , to track portions of whole items, we call them fractions. Specifically the denominator D instructs us to divide a whole item into D equal parts then we are to take N of these parts.

ND

Fractions provide answers for all equations of the form Ax + B = 0 (where A, B are integers)

However as we’ve seen that there is no fractional answer for x2 = 2 So we invented the symbol 2 ≈ 1.414.. to track this quantity and we call these numbers irrationals. We may represent all these numbers as decimals and we call all the decimal numbers the real numbers. BUT..

Complex NumbersThe equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative.

Complex NumbersThe equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.

Complex NumbersThe equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.

Complex NumbersThe equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

Use the square-root method.

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

Use the square-root method.x2 + 49 = 0x2 = –49

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

Use the square-root method.x2 + 49 = 0x2 = –49x = ±–49

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

Use the square-root method.x2 + 49 = 0x2 = –49x = ±–49 = ±49–1

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

Complex Numbers

We may write the solution of x2 = –r using i.Example A. Find the solutions of x2 + 49 = 0

Use the square-root method.x2 + 49 = 0x2 = –49x = ±–49 = ±49–1 = ±7i

The equation x2 = –1 does not have any real number solution since the square of any real number is always non-negative. We make up a new imaginary number –1to be a solution of this equation.We write i for –1 for simplicity.Hence, i2 = –1.

A complex number is a number of the form a ±biwhere a and b are real numbers,

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number.

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number.

Example B. 5 – 3i, 6i, –17 are complex numbers

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number.

Example B. 5 – 3i, 6i, –17 are complex numbers for 6i = 0 + 6i,

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number.

Example B. 5 – 3i, 6i, –17 are complex numbers for 6i = 0 + 6i, –17 = –17 + 0i.

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number.

Example B. 5 – 3i, 6i, –17 are complex numbers for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers.

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i)

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i)

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i

A complex number is a number of the form a ±biwhere a and b are real numbers, a is called the real part and ±bi is called the imaginary part of the complex number

Example B. 5 – 3i, 6i, –17 are complex numbers. for 6i = 0 + 6i, –17 = –17 + 0i.In particular, every real number a is also a complex because a = a + 0i.

Complex Numbers

(Addition and subtraction of complex numbers)Treat the "i" as a variable when adding or subtracting complex numbers. Example C.(7 + 4i) + (5 – 3i) = 7 + 4i + 5 – 3i = 12 + i(7 + 4i) – (5 – 3i) = 7 + 4i – 5 + 3i = 2 + 7i

(Multiplication of complex numbers)Complex Numbers

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i)

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The most important complex number multiplication formula is the product of a pair of conjugates numbers.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

Example E.

(4 – 3i)(4 + 3i)

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i)

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i) = (5)2 + 72

(Multiplication of complex numbers)To multiply complex numbers, use FOIL, then set i2 to be (-1) and simplify the result.

Complex Numbers

Example D. (4 – 3i)(2 + 7i) FOIL = 8 – 6i + 28i – 21i2 set i2 = (-1) = 8 – 6i + 28i + 21 = 29 + 22i

(Conjugate Multiplication)(a + bi)(a – bi) = a2 + b2 which is always a positive number.

The most important complex number multiplication formula is the product of a pair of conjugates numbers. The conjugate of (a + bi) is (a – bi) and vice versa.

Example E.

(4 – 3i)(4 + 3i) = 42 + 32 = 25(5 – 7i)(5 + 7i) = (5)2 + 72 = 54

Complex Numbers(Division of Complex Numbers)

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction,

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

(4 – 3i) (4 – 3i)

*

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 25

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–6

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,hence b2 – 4ac = (–2)2 – 4(2)(3) = –20.

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,hence b2 – 4ac = (–2)2 – 4(2)(3) = –20.Therefore

x = 2 ± –204

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,hence b2 – 4ac = (–2)2 – 4(2)(3) = –20.Therefore

x = 2 ± –204 = 2 ± 2–5

4

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,hence b2 – 4ac = (–2)2 – 4(2)(3) = –20.Therefore

x = 2 ± –204 = 2 ± 2–5

4 = 2(1 ± i5)4

Complex Numbers(Division of Complex Numbers)To divide complex numbers, we put the division as a fraction, then multiply the top and the bottom of the fraction by the conjugate of the denominator.

3 – 2i 4 + 3i Example F. Simplify

Multiply the conjugate of the denominator (4 – 3i) to the top and the bottom.(3 – 2i) (4 + 3i)

= (4 – 3i) (4 – 3i)

* 42 + 32 = 2512 – 8i – 9i + 6i2

–66 – 17i = 25

6 2517i –

Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.

To find b2 – 4ac first: a = 2, b = -2, c = 3,hence b2 – 4ac = (–2)2 – 4(2)(3) = –20.Therefore

x = 2 ± –204 = 2 ± 2–5

4 = 2(1 ± i5)4 = 1 ± i5

2

The powers of i go in cycle as shown below:

Complex Numbers

The powers of i go in cycle as shown below:

i

-1 = i2

Complex Numbers

The powers of i go in cycle as shown below:

i

-1 = i2

-i = i3

Complex Numbers

The powers of i go in cycle as shown below:

i

-1 = i2

-i = i3

1 = i4

Complex Numbers

The powers of i go in cycle as shown below:

i = i5

-1 = i2

-i = i3

1 = i4

Complex Numbers

The powers of i go in cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3

1 = i4

Complex Numbers

The powers of i go in cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4

Complex Numbers

The powers of i go in cycle as shown below:

i = i5

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3,

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3

Complex Numbers

The powers of i go in cycle as shown below:

i = i5 = i9 ..

-1 = i2 = i6 ..

-i = i3 = i7 ..

1 = i4 = i8 ..

Example H: Simplify i59

59 = 4*14 + 3, hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 = i3 = -i

Complex Numbers

Complex NumbersIn what sense are the complex numbers being numbers?Real numbers are physically measurable quantities (or the lack of such quantities in the case of the negative numbers).We know what it meaning to cut a stick with length of 3 ft but how do we make a stick of length 3i meters, or a cookie that weigh 5i oz? Well, we can’t. Complex numbers in general are not physically measurable in the traditional sense. Only the real numbers part of the complex numbers are tangible in the traditional sense. Complex numbers track both how much and in what direction. (hence the two–component form of the complex numbers).

Google the terms “complex numbers, 2D vectors” for further information.

Complex NumbersExercise A. Write the complex numbers in i’s. Combine the following expressions. 1. 2 – 2i + 3 + √–4 2. 4 – 5i – (4 – √–9) 3. 3 + 2i + (4 – i√5) 4. 4 – 2i + (–6 + i√3) 5. 4 – √–25 – (9 – √–

16) 6. 11 – 9i + (–7 + i√12) 7. ½ – (√–49)/3 – (3/4 – √–16) Exercise B. Do by inspection.8. (1 – 2i)(1 + 2i) 9. (1 + 3i)(1 – 3i) 10. (2 + 3i)(2 – 3i)11. (3 – 4i)(3 + 4i) 12. (9 + i√3)(9 – √3i) 13. (7 – i√5)(7 + i√5)14. (9 + i√3) (7 – i√5)(9 – i√3) (7 + i√5)15. (√3 + i√3) (√7 – i√5)(√3 – i√3)(√7 + i√5)Exercise C. Expand and simplify.16. (1 – 3i)(1 + 2i) 17. (2 + 3i)(1 – 3i) 18. (2 + 3i)(3 – 2i)19. (4 – 3i)(3 – 4i) 20. (5 + 3i)(5 + 3i) 21. (1 – i)2

22. (2 + 3i)2 23. (5 + 2i)2

Complex NumbersExercise D. Divide by rationalizing the denominators.

2 + 3ii24. 3 – 4i

i25. 3 – 4ii26.

1 + i1 – i27. 2 – i

3 – i28. 3 – 2i2 + i29.

2 + 3i2 – 3i30. 3 – 4i

3 – 2i31. 3 – 4i2 + 5i32.

33. Is there a difference between √4i and 2i?