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What is the square root of minus 1?
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The idea of the square root of a negative number is confusing at first, but it can be made easier by a few analogies. Forget about maths for a moment, and pretend that you are, for example, a simple sheep-farmer.
You have two sheep; someone owes you three sheep; they give you three sheep. You now have five sheep (2+3=5).
Suppose you have two sheep and you owe somebody else three sheep. As a farmer, you think �That�s impossible � I haven�t got three sheep to give away.� But the mathematician says �Never mind � let�s invent a new idea of negative numbers. There�s no such thing in the real world as a �minus sheep� or a �negative sheep�, but we can pretend that you have �minus one sheep�. By inventing a new type of number, we can deal with the situation more easily. So, in mathematical terms, the fact that you still owe the other person one sheep (in addition to the two which you have already given him) is expressed as �you own minus one sheep�. You don�t have to know what a �negative sheep� actually looks like.�
Now suppose that you own five sheep, and you want to leave them to your two sons when you die. As a farmer, you think �That�s impossible � I can�t give two-and-a-half sheep to each son, because I can�t cut a sheep in two pieces without reducing its value somewhat.� But the mathematician says �Never mind � let�s invent a new idea of fractional numbers. There�s no such thing in the real world as �half a sheep�, but we can pretend that your two sons will own �half� a sheep, and that two �halves� add up to one. By inventing a new type of number, we can deal with the situation more easily. So, in mathematical terms, the fact that your sons will share ownership of a sheep is expressed as �he owns half a sheep�. You don�t have to know what a �half sheep� actually looks like.�
You have two sheep; someone owes you three sheep; they give you three sheep. You now have five sheep (2+3=5).
Suppose you have two sheep and you owe somebody else three sheep. As a farmer, you think �That�s impossible � I haven�t got three sheep to give away.� But the mathematician says �Never mind � let�s invent a new idea of negative numbers. There�s no such thing in the real world as a �minus sheep� or a �negative sheep�, but we can pretend that you have �minus one sheep�. By inventing a new type of number, we can deal with the situation more easily. So, in mathematical terms, the fact that you still owe the other person one sheep (in addition to the two which you have already given him) is expressed as �you own minus one sheep�. You don�t have to know what a �negative sheep� actually looks like.�
Now suppose that you own five sheep, and you want to leave them to your two sons when you die. As a farmer, you think �That�s impossible � I can�t give two-and-a-half sheep to each son, because I can�t cut a sheep in two pieces without reducing its value somewhat.� But the mathematician says �Never mind � let�s invent a new idea of fractional numbers. There�s no such thing in the real world as �half a sheep�, but we can pretend that your two sons will own �half� a sheep, and that two �halves� add up to one. By inventing a new type of number, we can deal with the situation more easily. So, in mathematical terms, the fact that your sons will share ownership of a sheep is expressed as �he owns half a sheep�. You don�t have to know what a �half sheep� actually looks like.�
Now suppose you want to find the square root of a negative number. You think �That�s impossible � any number multiplied by itself is positive, so a negative number can�t have a square root.� But the mathematician says �Never mind � let�s invent a new idea of imaginary numbers. There�s no such thing in the real world the �square root of a negative number� but we can pretend that there is a type of number which is. By inventing a new type of number, we can deal with the situation more easily. So, in mathematical terms, let�s pretend that the square root of one is called �i�. You don�t have to know what �i� actually is; it just makes some calculations more easy.
Another way of thinking about it is in terms of operations rather than numbers.
Suppose that �multiply by minus one� means �turn something through 180 degrees�. In this way, an object which has been �multiplied by minus one� is a thing which has been rotated through 180 degrees, i.e. it is upside down. If you �multiply it by minus one� a second time, it will go through another 180 degree turn and will be back where it started. So multiplying something by minus one twice is the same as not doing anything. So minus one is the square root of one.
Similarly, �the square root of minus one� could be defined as �turn it through 90 degrees�.
If you turn something through 90 degrees, it will be lying on its side (i).
If you turn something through 180 degrees, it will be upside down (-1).
If you turn something through 270 degrees, it will be lying on its other side (-i).
If you turn something through 360 degrees, it will be where it started (i).
The concept of a number called �i� is weird, but it can be very useful for calculations by mathematicians, physicists, electricians, astronomers, and all sorts of cleverologists, as long as they don�t worry too much about what it actually is.
Another way of thinking about it is in terms of operations rather than numbers.
Suppose that �multiply by minus one� means �turn something through 180 degrees�. In this way, an object which has been �multiplied by minus one� is a thing which has been rotated through 180 degrees, i.e. it is upside down. If you �multiply it by minus one� a second time, it will go through another 180 degree turn and will be back where it started. So multiplying something by minus one twice is the same as not doing anything. So minus one is the square root of one.
Similarly, �the square root of minus one� could be defined as �turn it through 90 degrees�.
If you turn something through 90 degrees, it will be lying on its side (i).
If you turn something through 180 degrees, it will be upside down (-1).
If you turn something through 270 degrees, it will be lying on its other side (-i).
If you turn something through 360 degrees, it will be where it started (i).
The concept of a number called �i� is weird, but it can be very useful for calculations by mathematicians, physicists, electricians, astronomers, and all sorts of cleverologists, as long as they don�t worry too much about what it actually is.
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