Learning Outcomes

Express roots of an unfavorable numbers in regards to iExpress imaginary numbers together bi and complicated numbers as a+bi

You really require only one new number to begin working with the square root of negative numbers. That number is the square root of −1,sqrt-1. The actual numbers space those that have the right to be presented on a number line—they seem pretty real to us! once something is not real, we often say that is imaginary. So let us call this new number i and use that to represent the square source of −1.

You are watching: A complex number is a number of the form a + bi where


Because sqrtx,cdot ,sqrtx=x, us can also see that sqrt-1,cdot ,sqrt-1=-1 or i,cdot ,i=-1. We also know that i,cdot ,i=i^2, for this reason we have the right to conclude the i^2=-1.


The number i permits us to work-related with roots of all an adverse numbers, not just sqrt-1. There are two essential rules come remember: sqrt-1=i, and sqrtab=sqrtasqrtb. Girlfriend will use these rule to rewrite the square root of a negative number as the square source of a positive number time sqrt-1. Following you will certainly simplify the square root and also rewrite sqrt-1 together i. Permit us shot an example.


Simplify. sqrt-4

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Use the preeminence sqrtab=sqrtasqrtb come rewrite this as a product utilizing sqrt-1.

sqrt-4=sqrt4cdot -1=sqrt4sqrt-1

Since 4 is a perfect square (4=2^2), you have the right to simplify the square source of 4.


Use the meaning of i to rewrite sqrt-1 as i.


The price is 2i.

Use the dominion sqrtab=sqrtasqrtb come rewrite this together a product using sqrt-1.

sqrt-18=sqrt18cdot -1=sqrt18sqrt-1

Since 18 is not a perfect square, usage the same ascendancy to rewrite the using determinants that space perfect squares. In this case, 9 is the only perfect square factor, and the square source of 9 is 3.


Use the an interpretation of ns to rewrite sqrt-1 together i.


Remember to write i in prior of the radical.

The answer is 3isqrt<>2.

Use the preeminence sqrtab=sqrtasqrtb come rewrite this as a product utilizing sqrt-1.

-sqrt-72=-sqrt72cdot -1=-sqrt72sqrt-1

Since 72 is no a perfect square, usage the same ascendancy to rewrite the using factors that are perfect squares. Notice that 72 has actually three perfect squares as factors: 4, 9, and 36. The is easiest to usage the largest element that is a perfect square.


Use the meaning of ns to rewrite sqrt-1 together i.


Remember to write i in front of the radical.

The prize is -6isqrt<>2

A complex number is the sum of a genuine number and also an imaginary number. A complex number is express in standard form when written + bi where a is the real part and bi is the imagine part. Because that example, 5+2i is a complicated number. So, too, is 3+4isqrt3.

Imaginary numbers are differentiated from actual numbers since a squared imagine number to produce a negative real number. Recall, when a confident real number is squared, the result is a confident real number and also when a negative real number is squared, again, the an outcome is a hopeful real number. Facility numbers are a mix of real and also imaginary numbers. You can use the usual operations (addition, subtraction, multiplication, and so on) through imaginary numbers. You will certainly see an ext of the later.

Complex NumberReal PartImaginary Part
-frac35+isqrt2 -frac35 isqrt2
fracsqrt22-frac12i fracsqrt22-frac12i

In a number through a radical as part of b, such together -frac35+isqrt2 above, the imagine i need to be written in front of the radical. Though writing this number as -frac35+sqrt2i is technically correct, it renders it much more daunting to phone call whether i is inside or exterior of the radical. Placing it before the radical, together in -frac35+isqrt2, clears up any type of confusion. Look at these last 2 examples.

NumberComplex Form:a+biReal PartImaginary Part

By make b=0, any type of real number can be expressed together a complicated number. The real number a is composed as a+0i in complicated form. Similarly, any kind of imaginary number deserve to be expressed as a complicated number. By do a=0, any kind of imaginary number bi have the right to be written as 0+bi in complex form.


Write 83.6 together a complicated number.

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Remember that a facility number has actually the kind a+bi. You require to number out what a and b should be.


Since 83.6 is a real number, the is the real part (a) of the complicated number a+bi


A real number does no contain any imaginary parts, so the value of b is 0.

The price is 83.6+0i.

In the next video, we show more examples of how to create numbers as facility numbers.

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Complex numbers have actually the form a+bi, whereby a and also b are genuine numbers and i is the square root of −1. All genuine numbers deserve to be written as complicated numbers by setup b=0. Imagine numbers have the kind bi and can additionally be written as complicated numbers by setting a=0. Square root of negative numbers can be streamlined using sqrt-1=i and sqrtab=sqrtasqrtb.