If you are studying maths, you have probably heard of geometry, trigonometry, probability, decimals, equivalent fractions and algebra. If you are above primary school level, you would have of course been taught the maths basics of addition and subtraction, multiplication and division. But in all of your studies using numbers the most surprising maybe when letters of the alphabet or symbols are introduced into the equation. Like Pi, the golden ratio or proportion, x, y, e, i and the unique algebraic ‘numbers’ of maths.
Maths is such a vast subject to cover and learn, and it starts from basic math to extremely technical math concepts. As a student choosing to specialise in mathematics education will understand that to reach the core standard of the course that math practice will be a convenient addition to your life as a mathematician.
Have the skills to solve math sums and use your math skills actively to help you to memorise and improve your skills. If you find yourself struggling you can hire a tutor or create a study group with your fellow classmates. A math learning group at school has the common core benefit of allowing you to use math resources that are in your math class including textbooks, and your math teacher if they have time after class as well as joining peers who can work together to improve. This can help you with math questions, to solve problems, it will also make learning a bit more fun plus you benefit from free tutoring.
As you look at the long list of math problems and advanced math help that you need. Why not add the number i to the list. Hopefully not for too long as we take a look at it today.
What Is The Number I?
The need for i arose from the search for estimations and solutions to non-real equations such as third-degree equations. The number I thus is a concept allowing the mathematician to conceive a whole family of square roots in the negative number range.
The number i is so-called an imaginary number and complex number. It is called an imaginary number because you can not calculate the square root (multiply the value by itself) of negative numbers without producing a positive result. For example, 2² is 4, just like (-2) ².
In mathematics, some equations have no real solution because there is no number whose square root is negative. Therefore complex numbers create the foundation to mix real numbers and imaginary numbers together, solving these unsolvable problems. Complex numbers are equations which use the imaginary number i along with real numbers to calculate and solve the unsolvable.
To reinforce this mathematically, we must go back to middle school math class where we learn the basic math rules of the signs such as multiply, subtract, divide or add.
- Imaginary number i looks like this: i, √-1 or j
- Complex numbers look like this: (a+bi)
- Where a = a real number
- Where bi = an imaginary number
- Real numbers and whole numbers look like this: 1, 2, 3
- Negative numbers look like this: -1, -2, -3
In the case of real numbers and imaginary numbers, only numbers from the same family can mix. So, for example, this is a simple, complex number equation.
- (3+2i)+(-1+i) real numbers are in bold font
- Simplified by 1 level is (3+(-1)) + (2i+i)
- Where the calculation begins and the real numbers are brought together to be calculated, and the imaginary numbers go together to be calculated.
- The solution to the top equation is complex number 2+3i
- This can now be plotted on a graph if you decide graphing is the way that you want to display your complex number.
Why The Letter (Or Number) I?
Ever wondered why this non-determined number was given the letter I?
The letter I, or i, is used in Maths to signify that a number is an imaginary number. Now, every time you see this in an equation, you'll be reminded that the number is a mysterious, imaginary digit and this could well help you to get around the formula and clarify in your mind what the task or question is asking you to do!
The Birth Of The Imaginary Number
I, or the imaginary number, is more complex than it looks.
Wikipedia reports that: "The number can be written as a real number multiplied by the imaginary unit i, [...] which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary."
The concept came about in the 17th century and became widely accepted following the study and research work of Leonhard Euler and Carl Friedrich Gauss.
"An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number. Some authors use the term pure imaginary number to denote what is called here an imaginary number, and imaginary number to denote any complex number with non-zero imaginary part.
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.
Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as negative numbers, were poorly understood and regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory. The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818)."
The History Of I In Pre Algebra Continued
In the 16th century, Girolamo Cardano and Jerome Cardan set out to solve a third-degree equation √-15 And came upon impossible numbers. But Raphael Bombelli is the first mathematician to have delved into the mathematical concept and elaborate on the calculation rules. This is where complex number begin to appear in history, with algebra and impossible numbers.
- Einstein, the beloved and famous mathematician, used the impossible number and complex number i to build his theory of relativity.
- Rene Descartes, the French philosopher, named the numbers imaginary numbers.
- Leonhard Euler, the co-inventor of the number e (the exponential/infinite number) works with the number i to expand his work on the impossible and imaginary.
- During the 19th century, the number i with the help of authors and mathematicians like CF Gauss. The imaginary number i ends up being accepted as numbers in its own right.
- The number i is used in the common core standards of computation.
Why Use Imaginary Numbers?
It is true that imaginary numbers, complex numbers, trig. and algebra. Have their place in mathematics, and that their application may not be useful for someone who does not study maths. It is also true that this is not the kind of maths that you will find in the kindergarten or primary level classroom.
However, by the time you get to the high school math classroom, there you will find it, discrete and waiting for you. Along with linear equations, differential equations, Pythagorean theorem, triangles, Geometry, Trigonometry, Probability and the mastery of other mathematical concepts.
Oh yes! High school is no joke, and at this grade level, the equations become advanced. This theorem will be quite challenging for students who still struggle with arithmetic, subtracting, multiplication problems or average grade math.
As a learner of maths, learning about the number I, should be added to your list of math skills as the use and application of complex numbers are multiple.
However, the learning about complex numbers doesn’t have to be boring, learning fun can go along with even the most challenging maths.
How To Explain Imaginary Numbers To Kids
Trying to explain to a kid that there is a number in existence that isn't really a number, but is at the same time every number, is an extremely difficult concept. So, if you are the parent of a child trying to grasp pre-algebra ahead of starting the more complicated school courses, then you might like some tips on how to approach this imaginary number!
Top tips for teaching children about the letter I:
- Focus on relationships, rather than studying mechanical formulas too closely.
- Talk about complex numbers like they are an upgrade to the number system we are used. Think back to when they ventured away from numbers 1 to 100 and then grasped zero, decimals and negative numbers. This is the next step.
- Use visual diagrams wherever possible, not just text, to understand the idea as an imaginary number calls for the use of imagination and not just intellect.
Ways Math Fun Can Help Adults Study Complex Numbers
When you are relaxed and have fun, you learn easier, retain knowledge for longer and not to mention the best part you enjoy the process. Cool math games were very popular to learn your times tables, calculus adding and subtracting when you were in primary school. Learning fun with songs, puzzles, perhaps a number jigsaw. At this age, you've played, and it became interactive math.
The math curriculum gave you a number system, counting, Rounding up or down was easy, decimal places, fractions came naturally, then it starts to get more challenging, with number theory and we stop playing. Often we also stop enjoying the math process too. Just because the curriculum becomes difficult doesn’t mean that you shouldn’t enjoy learning about maths. Playing a bit each day can help you to excel in your grade level or even surpass the core standards. Remember not all games for kids are exclusively for kids.
Fun Math Study Tips
- YouTube videos: watching math videos as a part of your online math study can be really engaging and give you a nice break from the books. Often when you watch a few videos, you will find a math teacher or math tutor online who explains thing since the way that it clicks in your mind. Helping to bring logic and number sense you what you are studying. Before long it will be as easy as telling time.
- Create lesson plans: Making your own study program that splits up the parts that you are struggling with will make it easy to take in. Bite-sized study sessions that are interactive math study sessions with help you to retain what you have learnt. You can use math worksheets, get creative and you’re your own number system, do some puzzles, quizzes…etc.
- Know what you know: review the parts that you have understood quickly every day to refresh and build your self-esteem, and check the math facts as you know
- Play online math games: free online math apps are a great way to practice what you know and open your mind up to take in new knowledge.
- If you have a tutor or are a part of a study group be open about where you feel that you need help.
- You can take a specific math course in the area that you need more help in.
- Games: Sudoku is an excellent math game from Japan that is a terrific exercise for the brain.
- Sing the multiplication tables: They used to sing as songs in the classroom to help with their memorisation. You can do the same for the math facts of complex numbers.
- Ask for help: leave a note for teachers who tend to be quite creative in their teaching style and ask for tips for maximising your learning.
Have fun with your mathematics, even if your personal study doesn’t follow the curriculum or state standards. You have to do what is best for you to perform at your personal best. Place value on learning these new numbers, they will be a foundational knowledge as your maths work gets steadily more challenging.
Study Imaginary And Complex Numbers
If you are particularly interested in discovering further maths and finding out about imaginary and complex numbers, you can educate yourself with lessons. Your first port of call, if you are seriously looking for education offering you a qualification at the end of it, is to search for Maths courses at a university or college. You may choose to consult establishments that are local to you, especially if this is something you see yourself doing as further study alongside a job or existing course. Here is just one example of undergraduate Maths course in the south.
The University of Essex
The University of Essex offers a BSc in Mathematics, an undergraduate course, which teaches maths lovers a range of further maths skills to help them progress with a career in related fields.
See the course overview below:
"Mathematics is the language that underpins the rest of science. At Essex, Mathematics has truly broad reach; we are working on projects ranging from the economic impact of the behaviour of dairy cows, to understanding crowd behaviour through modelling a zombie apocalypse, to circular Sudoku and other puzzles. Our Department of Mathematical Sciences is genuinely innovative and student-focused.
On our BSc Mathematics you study a wide range of topics including:
- Finance and Big Data
- Discrete mathematics, languages and semigroup theory
- Probability and applied statistics
- Bioinformatics and mathematical ecology
Our interdisciplinary research recognises that mathematics, including what can be very abstract mathematics, is an essential part of research in many other disciplines.
You therefore can gain an exceptional range of knowledge and skills that are currently in demand in mathematically oriented employment; in business, commerce, industry, government service, education and in the wider economy.
This programme will meet the educational requirements of the Chartered Mathematician designation, awarded by the Institute of Mathematics and its Applications, when it is followed by subsequent training and experience in employment to obtain equivalent competences to those specified by the Quality Assurance Agency (QAA) for taught masters degrees."
Aren't sure if you qualify to attend this type of degree course? Check out the entry requirements below:
"UK entry requirements
A-levels: BBB, including Mathematics
Please note we are unable to accept A-level Use of Mathematics in place of A-level Mathematics
IB: 30 points, including Higher Level Mathematics grade 5. We are also happy to consider a combination of separate IB Diploma Programmes at both Higher and Standard Level.
Exact offer levels will vary depending on the range of subjects being taken at higher and standard level, and the course applied for. Please contact the Undergraduate Admissions Office for more information."
The Open University
One of the most convenient ways to get a qualification in further Geometry is to look at enrolling on courses at the Open University.
An Introduction to Complex Numbers, a free course offered by the Open University, teaches you how complex numbers are defined, encourages you to examine their geometric representation and then gets you to look at the methods for finding the nth roots of complex numbers and the solutions to simple polynominal equations.
"After studying this course, you should be able to:
- perform basic algebraic manipulation with complex numbers
- understand the geometric interpretation of complex numbers
- know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations."
At no cost at all, it seems like a no brainer to go ahead and make the most of this course!
If you can't commit enough time to an official course, then you might like to opt for a self-teaching method. The Internet is a fabulous thing for students as it offers almost endless options to learn or acquire new knowledge.
Just by typing in a quick search on Google, we have found a range of video tutorials on YouTube which explain to Maths students about complex numbers and imaginary numbers.
Khan Academy is one of the top education companies offering online training, and they don't limit themselves to YouTube. There is also a website and an app, perfect for all ages and levels. Though primarily aimed at Physics students, the company also covers a range of complicated maths concepts and is a popular choice among learners:
"All of the nonprofit's site's resources are available to anyone, and you can access its contents using a variety of different platforms. Take part in the interactive classroom blackboard teaching style with short quizzes and questions to test your knowledge and improvement. If you really love Khan Academy and what it offers you in terms of Physics help (it also offers help in other subjects such as Maths, Science and Computing) then you can donate to support its ongoing existence."
The advantage of having a private maths tutor is that you have the full, undivided attention of your teacher, and they can tailor the work to your needs. So if formulas are an area of weakness for you, your tutor can place emphasis on this in your sessions and homework studies.
Superprof, a leading platform for tutors and students to connect and form working partnerships, offers a user-friendly website on which you can instantly locate tutors offering maths tutoring services in your area, as well as those who are able to offer remote online tuition.
There are close to 60,000 tutors listed online who can help you electronically or via video call with your maths concerns. With prices starting from just £5 per hour, you can find someone who can meet your needs and get you moving forward with your math lessons. Some are mathematicians, some are qualified teachers, whilst others are individuals who are talented with numbers and want to pass on their knowledge and skills.
So, if you are still finding it hard understanding the nature of the letter I, then go ahead and contact prospective tutors in order to get learning!