In the penultimate article of his series, Dexter Booth defines what we mean by an ‘imaginary’ number
The rational and irrational numbers are collectively called the real numbers. Why real? It’s another of those words that seem to come from nowhere and I shall explain the reason for the word shortly. For now, we shall re-cap. We have a collection of numbers that only exist in the mind but we do have a consistent means of representing them symbolically – of communicating our ideas about them from one mind to another. We can manipulate the symbols to demonstrate adding, subtracting, multiplying and dividing them and we can also raise them to a power. The real numbers are closed under the first four arithmetic operations; the addition, subtraction or multiplication of two real numbers is another real number. The division of two real numbers is another real number but be careful here, there is a caveat. We cannot divide by zero because we cannot define the outcome. There is nothing we can do about this restriction – we must just accept it. If we try a ‘what if?’ we enter the realm of that notoriously misunderstood word infinity and that is a whole other blog post.
If a real number is raised to a real power the result is another real number.
Really? Let’s look at this a little more closely. Consider the product
This is how powers add together. Taking this further we see that:
The square root of 4 being that number which when multiplied by itself results in 4. Now there are two such square roots of 4 because:
We write this as:
Which reads as the square root of 4 is plus or minus 2. And now, using a calculator we find that
The calculator is telling us that there is no real number that equals the square root of minus 4. Therefore if a real number is raised to a real power the result may not be another real number
Think about this one. If you multiply two positive numbers together you obtain a positive number. If you multiply two negative numbers together you again obtain a positive number. You only obtain a negative number if you multiply a negative number and a positive number. So there is no single number which when multiplied by itself gives a negative result. Consequently, we can say unequivocally that there is no real number that is equal to the square root of a negative real number, so the real numbers are not closed under the operation of raising to a power. The positive real numbers are closed but the negative real numbers are not. However, there is nothing intrinsically wrong with the arrangement of symbols in
We could discount this arrangement of symbols but it would seem to be rather arbitrary to do so and we don’t like these seemingly arbitrary rules. I feel a ‘what if?’ coming on. Could we invent a number that was equal to
And if so, how would it behave? We can write:
So every square root of a negative real number can be reduced to a real number multiplied by minus one to the power one-half:
And it is this arrangement of symbols that represents what is at the heart of the problem and what is defined to be a new type of number. It was originally called an impossible number because we can neither count nor measure with it so it was thought to be impossible to use and was just a curiosity. However, as we have just seen we can neither count nor measure with irrational numbers either. In the 17th century Rene Descartes was the first to call it imaginary but only because he felt it was of no value. However, over time the name stuck and that was the reason why our earlier numbers were called real numbers so as to distinguish them from the imaginary numbers. It really is rather silly to call them imaginary numbers because it tends to ascribe to them an illusory nature, further compounded by using the word real for our other numbers.
“Huh, so 15 is a real number and [−1]1⁄2 isn’t really a number at all.”
Such is the power of a label and the subtlety of the human mind to convert ‘not a real number’ into ‘not really a number’. In fact, as we have said many times before, all numbers are just inventions of the mind so it is really confusing to call one type real and the other type imaginary. We could equally call them Type 1 and Type 2 numbers but human beings are susceptible to historical precedence so we are stuck with calling them real and imaginary numbers. As Charles Dickens said ‘the wisdom of our ancestors is in the simile; and my unhallowed hands shall not disturb it, or the Country’s done for.’ †
The notation for what is called the imaginary unit is
Where the surd notation (√) indicates the positive square root. So now I have probably offended everyone. Mathematicians and physicists use i as the imaginary unit whereas engineers use j because i is already used to stand for current measured in amperes. Whether we use i or j does not matter here. What does matter is that we have an entirely new type of number, spawned directly from our symbolism for real numbers. Can we manipulate them as we can our real numbers? We can. We can add, subtract, multiply and divide them and they are closed under addition and subtraction but not under multiplication or division. For example,
Both of the latter represent real numbers, not imaginary numbers. The next natural question to ask is can we raise an imaginary number to an imaginary power? Not an easy question to answer properly at this stage in the process. The bald answer is yes, after all there is nothing inherently incorrect, nothing expressly forbidden about, for example, the symbol structure of the imaginary unit raised to the power of the imaginary unit:
Indeed this arrangement of symbols represents a real number with the decimal form of:
Bet you weren’t expecting that! We seem to be jumping from numbers Type 2 to numbers Type 1 – multiplying and dividing imaginary numbers and raising an imaginary number to an imaginary power and each time producing a real number. There seems to be a link between the two types of number. Indeed there is.
†A Christmas Carol