ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Complex Numbers: Why √(-1) = i Is A Misleading Starting Point For Understanding Complex Numbers

Updated on May 29, 2014

Understanding The Number Line

The way complex number concept is introduced misleads students.

Most people are familiar with the number line and most ancient tribes had an idea of counting. The base-ten comes as a natural counting process for most tribes. The reason can be as simple as the fact with ten digits (fingers). Some animals have a way of knowing that one or two members are missing.

The straight number line that stretches from minus-inifinity to plus-plus infinity is well understood to most by second grade. Before that most people can count from to infinity through stages from ten, hundred, thousand depending on thier abilities. Negative numbers concept may a little bit difficult to grasp for many before first grade. It does not take long to understand that the number line starts before zero. Teachers can use the concept of borrowing and lending as a pedagogy.

"If you borrow Susan's orange today, you have minus one orange, when you bring two oranges tomorow you will only be able to eat one orange because you need to give back Susan's orange from the two." Something like this is a good teaching aid.

What is not easy to conceptualize is the concept of a complex plane. The situation is not made easier when people are introduced to this concept through the assertion: √(-1) = i .

While √(-1) = i is a true statement, it is only an instant of the whole concept. A better introduction to the complex plane is preferable if science and engineering students will have a lasting appreciation.

This article discusses complex number fundamentals and what they mean to engineers.

Operator Concept: √(-1) = i

The strength of mathematics comes from its abstract nature, and therein comes its weakness. This is a strong statement.

The introduction of complex numbers as √(-1) = i is adequate from a pure mathematics point of view. Science and engineering students are better served if complex numbers are introduced with i , ( the √(-1) ), as an operator that shifts the phase of a number by 90-degrees. A second operation would result in a 180-degree shift. This explains why (i)2 = -1 . This explanation is not only practical, but is even a better abstraction than √(-1) = i.

Algebraically the two statements are the equivalent.

To say , √(-1) = i is a correct statement but it does not explain what is taking place, it is more like explaining a concept through the most obvious example. Or worse by giving the consequence as an explanation of the cause.

A better understanding of the complex number plane will help understanding natural phenomenon like echos, colour spectrum, sound quality and image processing, to name but just a few.

The Complex Number Plane

Real numbers are a subset of complex numbers. Human beings (and other animals) were using complex numbers before they understood the mathematics. A bat using echo-location applies the concept of phase shift to calcuate distances.

The very concept of interest rates, that money today is not the same as money tomorrow carries a deeper understanding of complex numbers than people to accept. There is a time difference on the arrival of the money which can be expressed as a phase shift.

Animals have abilities to sense minute differences in sound, color, distances, speeds and acceleration than any other mechanical or electronic sensor ever manufactured. Our ears, eyes and fingers can pick up differences in time of arrivals of signals to make sense of where a sound is coming from or how far one object is from another. Or to be able to track and chase a target.

The above paragraph explains why humans are still the best machines out there. Machines are getting advanced partly due to development of algorithms . These algorithms must be able to mimic how we sense and calculate natural occurrences.

To understand complex numbers, a scientist must abstract a concept of numbers spread out on a plane rather than just on a straight line. Numbers on a complex plane have both magnitude and phase differences.

Comments

    0 of 8192 characters used
    Post Comment

    • mgt28 profile imageAUTHOR

      mgt28 

      4 years ago

      Hackslap,

      I can understand, Complex Numbers is a useful and interesting branch of Mathematics.

    • Hackslap profile image

      Harry 

      4 years ago from Sydney, Australia

      All I'd say here is .. this was a trip down memory lane to 1999 when I was studying complex numbers during year 11 at school ...and confused me to no end lol

      Great article ..perhaps now at 31 it might make all sense lol

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://hubpages.com/privacy-policy#gdpr

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)