What Is A Bi Form

Complex Numbers

Learning Objective(s)

· Express roots of negative numbers in terms of i.

· Limited imaginary numbers as bi and complex numbers as a + bi.

Introduction

Several times in your learning of mathematics, you accept been introduced to new kinds of numbers. Each fourth dimension, these numbers made possible something that seemed impossible! Earlier y'all learned about negative numbers, yous couldn't subtract a greater number from a bottom one, but negative numbers give us a fashion to do it. When you were learning to dissever, you lot initially weren't able to do a problem like thirteen divided by 5 because 13 isn't a multiple of 5. You then learned how to do this problem writing the answer every bit 2 remainder 3. Somewhen, y'all were able to express this answer as . Using fractions immune you to make sense of this division.

Up to at present, you've known it was impossible to take a square root of a negative number. This is truthful, using just the real numbers. But hither yous will learn about a new kind of number that lets you lot work with square roots of negative numbers! Like fractions and negative numbers, this new kind of number will let you practise what was previously incommunicable.

Using i to Simplify Roots of Negative Numbers

You lot really need only one new number to start working with the square roots of negative numbers. That number is the square root of − 1, . The existent numbers are those that can be shown on a number line—they seem pretty real to us! When something'south non real, you oftentimes say it is imaginary. So allow's call this new number i and use it to stand for the square root of −1.

Because , we tin can also run across that  or . Nosotros also know that , so we can conclude that .

The number i allows us to work with roots of all negative numbers, not just . There are ii important rules to remember: , and . You volition use these rules to rewrite the square root of a negative number every bit the square root of a positive number times . Next you volition simplify the square root and rewrite  equally i. Let's try an instance.

Example
Problem	Simplify.
		Utilise the rule  to rewrite this as a product using .
		Since 4 is a perfect square (four = ii2), you lot can simplify the square root of 4.
		Employ the definition of i to rewrite  equally i.
Answer

Instance
Problem	Simplify.
		Employ the rule  to rewrite this as a product using .
		Since 18 is non a perfect foursquare, use the same dominion to rewrite it using factors that are perfect squares. In this case, 9 is the only perfect square factor, and the square root of 9 is 3.
		Use the definition of i to rewrite  as i. Remember to write i in front of the radical.
Reply

Case
Problem	Simplify.
		Use the dominion  to rewrite this as a product using .
		Since 72 is not a perfect foursquare, use the same rule to rewrite it using factors that are perfect squares. Observe that 72 has three perfect squares equally factors: 4, 9, and 36. It's easiest to use the largest factor that is a perfect square.
		Use the definition of i to rewrite  as i. Retrieve to write i in forepart of the radical.
Answer

Yous may accept wanted to simplify  using different factors. Some may have idea of rewriting this radical equally , or , or  for case. Each of these radicals would have somewhen yielded the same answer of .

Rewriting the Foursquare Root of a Negative Number

· Notice perfect squares within the radical.

· Rewrite the radical using the rule .

· Rewrite  every bit i.

Case:

Simplify.

A) 5

B)

C) vi

D)

Evidence/Hide Answer

A) 5

Incorrect. Y'all may have noticed the perfect square 25 as a cistron of 50, merely forgot the balance of the number under the radical. The correct respond is:

B)

Incorrect. While , the negative under the radical cannot be moved outside

the radical. Remember, . The right answer is:

C) 5i

Incorrect. You may take correctly noticed the perfect square 25 equally a factor of 50, and correctly used , but you lot forgot the remaining factor of −50, which is 2. The correct answer is:

D)

Right.

Imaginary and Complex Numbers

You can create other numbers past multiplying i by a real number. An imaginary number is any number of the form bi, where b is real (just non 0) and i is the square root of −1. Look at the following examples, and discover that b tin be any kind of real number (positive, negative, whole number, rational, or irrational), simply not 0. (If b is 0, 0i would just exist 0, a real number.)

Imaginary Numbers
3i (b = iii)	− 672i (b = − 672)
(b = )	(b = )

Y'all can use the usual operations (addition, subtraction, multiplication, and so on) with imaginary numbers. You'll see more of that, later on. When y'all add together a real number to an imaginary number, however, you lot go a circuitous number. A circuitous number is whatever number in the form a + bi, where a is a existent number and bi is an imaginary number. The number a is sometimes called the existent office of the complex number, and bi is sometimes called the imaginary function.

Complex Number	Real part	Imaginary function
3 + viii	iii	7i
18 – 32i	18	−32i

In a number with a radical equally part of b, such as  above, the imaginary i should be written in front end of the radical. Though writing this number as  is technically correct, it makes information technology much more difficult to tell whether i is inside or outside of the radical. Putting it earlier the radical, as in , clears up any confusion. Wait at these last two examples.

Number	Number in complex form: a + bi	Real part	Imaginary function
17	17 + 0i	17	0i
−iiii	0 – 3i	0	−iiii

By making b = 0, whatever existent number can be expressed as a complex number. The real number a is written a + 0i in complex form. Similarly, any imaginary number can exist expressed as a complex number. By making a = 0, any imaginary number bi is written

0 + bi in circuitous course.

Example
Problem	Write 83.half dozen as a complex number.
	a + bi 83.half-dozen + bi	Recollect that a complex number has the form a + bi. You need to figure out what a and b need to be. Since 83.6 is a real number, information technology is the real function (a) of the complex number a + bi. A real number does not contain any imaginary parts, so the value of b is 0.
Answer	83.half-dozen + 0i

Example
Problem	Write −threei as a complex number.
	a + bi a – iiii	Recall that a complex number has the form a + bi. Y'all need to figure out what a and b need to be. Since −3i is an imaginary number, it is the imaginary part (bi) of the complex number a + bi. This imaginary number has no real parts, so the value of a is 0.
Answer	0 – 3i

Which is the existent role of the complex number −35 + 9i?

A) 9

B) −35

C) 35

D) 9 and −35

Show/Hibernate Answer

A) 9

Wrong. The number 9 is in the imaginary office (9i) of this complex number. In a complex number a + bi, the real part is a. In this case, a = −35, then the existent part is −35.

B) −35

Correct. In a circuitous number a + bi, the existent part is a. In this example, a = −35, and so the real role is −35.

C) 35

Wrong. In a complex number a + bi, the real part is a. In this example, a = −35, so the real role is −35. The real part can exist any real number, including negative numbers.

D) 9 and −35

Wrong. The number 9 is in the imaginary function (ninei) of this complex number. In a complex number a + bi, the real part is a. In this case, a = −35, so the real function is merely −35.

Summary

Complex numbers have the form a + bi, where a and b are real numbers and i is the square root of −i. All real numbers can be written equally complex numbers by setting b = 0. Imaginary numbers have the form bi and tin besides exist written as complex numbers by setting a = 0. Foursquare roots of negative numbers can be simplified using  and

What Is A Bi Form,

Source: http://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U16_L4_T1_text_final.html

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