For many years, mathematicians avoided using or thinking about the square root of negative numbers. Preferring to think in terms of real numbers, they simply agreed with each other that roots of negative numbers made no sense at all (for similar reasons, negative numbers were avoided for many centuries). After all, there is no physical length that can possibly equal such a number. What possible use or relevance could it have for our world?
The Italian mathematician Gerolamo Cardano (1501-1576) was the first to have explored complex numbers seriously. He used them in order to find solutions to cubic equations … and called them “fictitious.” Little did he realise what an incredible explosion of understanding would result from his first foray into this field!
It transpires that complex numbers are intimately connected with exponential equations and with trigonometry (see the image to the left, where i represents √(-1) and is called an “imaginary number”).
Nowadays, complex numbers are used whenever any repetitive, cyclic, or wave motion is being analysed (from star light to quantum mechanics to electric tuning circuits to shock waves during earthquakes)! Not only do mathematicians study them for the pure joy (and fascination) of the exercise, but they are used in many disciplines such as physics, chemistry, biology, engineering, statistics and economics.
Briefly, an imaginary number is one that is obtained by taking the square root of a negative number. If we define √(-1) as i (meaning imaginary), all imaginary numbers can be written as the product of i and some real number. This is because √(-k) = √(-1)⋅√k = i√k, where √k is real.
Complex numbers are hybrid numbers, obtained when we add a real number and an imaginary number together … such as 2 + 3i.
There is much, much more … so, watch this space!
We were all deeply impressed with Graeme from the very beginning. [Lucas’] confidence radically changed within only 3 weeks of Graeme’s instruction and assistance. I am sure this was largely attributed to Graeme’s infectious passion and love for his subject, and his high personal level of skill, teaching experience and understanding of mathematics. Lucas always found Graeme could explain concepts so knowledgeably and easily, which really did transform Lucas’ appreciation and enjoyment of the subject. Graeme always gave Lucas assistance and time over and above his hour’s tutoring session. Graeme has developed a wonderful relationship and more of a mentoring role which has been of great assistance and value to Lucas as he faces the pressures of his final year of school. I have no hesitation in recommending Graeme as an outstanding tutor and friend, and a very wise, knowledgeable and capable teacher of the highest calibre.
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