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Can someone show me how to write complex numbers in standard form? I missed a few days of class and do not have the text book. Answering a simple question like the one below would help

Write the complex number in standard form. $6 + \sqrt{−16}$

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    I hope that you get the textbook soon.2012-05-29
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    Have you tried talking to your teacher, or your fellow students? Those should be your first recourses when making up missed classes.2012-05-29
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    This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level.2017-06-15
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    All, this OP also posted [this earlier version](https://math.stackexchange.com/q/151017/11619), In the same vein. Back in the day this question escaped closure. Nowadays it wouldn't stand a chance. I don't understand the votes to reopen at all???2018-05-04
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    @JyrkiLahtonen, one of the votes to reopen is explained in one of the answers here: https://math.meta.stackexchange.com/questions/28348/downvote-close-delete2018-05-05

2 Answers 2

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$\sqrt{-1}$ is written as just $i$ for "imaginary".

$\sqrt{-x}$ can be factored as $\sqrt{-1\vphantom{x}}\sqrt{x} = i\sqrt{x}$.

"Standard form" for complex numbers is $a + bi$ where $a$ and $b$ are real numbers. If $a$ or $b$ is 0, you omit that part. For example, you write $3 + 0i$ as just $3$, and $0 + 3i$ as just $3i$.

For your example, you have $6+\sqrt{-16} = 6 + i\sqrt{16} = 6 + 4i$. The "standard form" is $6+4i$.

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    You have to be carefull, when defining the imaginary unit as you did, see https://en.wikipedia.org/wiki/Imaginary_unit#Proper_use - the correct way is, to say that \(i\) is defined by \(i^2 = -1\), which is note the same as your definition (to be precise, the root is not defined for complex numbers).2013-12-16
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Note that $i^2=-1$, so $\sqrt{-16}=\sqrt{i^216}=i4. $ Hence $6+\sqrt {-16}=6+i4{}$

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    Who voted this down? Can you explain why?2012-05-29
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    I voted it down because the OP's teacher is likely to reject "$6+i4$" as a "standard form" for complex numbers.2012-05-29
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    @MarkDominus. So what is the standard form? $6+4i$?2012-05-29
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    @Mark: $a+ib$ and $a+bi$ are both explicitly mentioned as standard form in some textbooks, and regardless I don't think that a teacher would be reasonable to not accept $6+i4$ as standard form.2012-05-29
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    @MarkDominus. $4i$ or $i4$ does not make any difference. You dont vote down when you dont know. You should find out before you vote down.2012-05-29
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    My apologies. ${}$2012-05-29
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    $a+ib$ is fine. $6+i4$ is bizarre.2012-05-29
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    A pedantic (at this level, at least) note: square roots can do funny things with $i$. For example, the square root of $i^4 = 1$ is $1$, right? And yet it also seems reasonable to write $\sqrt{i^4} = \sqrt{(i^2)^2} = i^2 = -1$.2012-05-29
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    @GerryMyerson. It is not bizarre provided you know what it means. The reason why some textbooks prefer this style is that in case of squre roots the meaning can be misleading. For example $i\sqrt{3}$ might be preferred over $\sqrt{3}i$. This is because the latter appears as if one is finding the square root of the product of $3$ and $i$.2012-05-29
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    $i\sqrt3$ is fine. $i4$ is bizarre. It looks like someone meant $i_4$ but couldn't find the underscore key.2012-05-29