If the product of the two numbers is positive, they must have the same sign, and since their sum is positive, this means they must both be positive. It was a disturbing discovery. But he was surprised to find that they obeyed many of the same rules that real numbers do.
Complex numbers are made up of a real part and an imaginary part. To add and subtract complex numbers, you just combine the real parts and the imaginary parts, like this:. Is this really a complex number, or is it something else? So we can multiply complex numbers, but how do we divide them?
The key is understanding the relationship between division and multiplication. I often tell students that there is no such thing as division: There is only multiplication by the reciprocal.
And it may be a little surprising that this number is — i! This means that if we ever want to divide a number by i , we can just multiply it by — i instead. For other complex numbers, the arithmetic may get a little harder, but the reciprocal idea still works.
The product of the complex number and its conjugate is a real number! This property of conjugates helps us compute the reciprocal of any complex number. The introduction of this one new non-real number — i , the imaginary unit — launched an entirely new mathematical world to explore. It is a strange world, where squares can be negative, but one whose structure is very similar to the real numbers we are so familiar with.
And this extension to the real numbers was just the beginning. The quaternions are structured like the complex numbers, but with additional square roots of —1, which Hamilton called j and k. For instance, will the system be closed under multiplication?
Will we be able to divide? Hamilton himself struggled to understand this product, and when the moment of inspiration finally came, he carved his insight into the stone of the bridge he was crossing:. People from all over world still visit Broome Bridge in Dublin to share in this moment of mathematical discovery. The other products can be derived in a similar way, and so we get a multiplication table of imaginary units that looks like this:. Notice this means that, unlike with the real and complex numbers, multiplication of quaternions is not commutative.
Multiplying two quaternions in different orders may produce different results! To get the kind of structure we want in the quaternions, we have to abandon the commutativity of multiplication. This is a real loss: Commutativity is a kind of algebraic symmetry, and symmetry is always a useful property in mathematical structures. But with these relationships in place, we gain a system where we can add, subtract, multiply and divide much as we did with complex numbers.
To add and subtract quaternions, we collect like terms as before. To multiply we still use the distributive property: It just requires a little more distributing. And to divide quaternions, we still use the idea of the conjugate to find the reciprocal, because just as with complex numbers, the product of any quaternion with its conjugate is a real number. Thus, the quaternions are an extension of the complex numbers where we can add, subtract, multiply and divide.
And like the complex numbers, the quaternions are surprisingly useful: They can be used to model the rotation of three-dimensional space, which makes them invaluable in rendering digital landscapes and spherical video, and in positioning and orienting objects like spaceships and cellphones in our three-dimensional world.
Negative numbers do not have square roots — there is no number that, when multiplied by itself, gives a negative number. Tartaglia and his rival, Gerolamo Cardano, observed that, if they allowed negative square roots in their calculations, they could still give valid numerical answers Real numbers, as mathematicians call them.
Mathematicians use i to represent the square root of minus one. We can use it to find the square roots of negative numbers though. This means that the square root of -4 is the square root of 4 multiplied by the square root of In symbols:.
The square root of 4 is 2, and the square root of -1 is i , giving us the answer that the square root of -4 is 2 i. We should also note that -2 is also a square root of 4 for the reasons stated above. This means that the square roots of -4 are 2 i and -2 i. The arithmetic of i itself initially posed an obstacle for mathematicians.
I stated above that a negative times a negative gives a positive and we are innately familiar with the idea that a positive times a positive gives a positive.
With the imaginary unit, this seems to break down, with two positives multiplying to give a negative:. This was a problem for some time and made some people feel that using them in formal mathematics was not rigorous.
Rafael Bombelli, another Italian renaissance man, wrote a book called, simply, Algebra in where he tried to explain mathematics to people without degree-level expertise, making him an early educational pioneer. The work of these mathematicians on imaginary numbers allowed the development of what is now called the Fundamental Theorem of Algebra.
In basic terms, the number of solutions to an equation is always equal to the highest power of the unknown in the equation. The highest and only power of the unknown x in the equation is two, and lo and behold we found two answers, 2 i and -2 i. With a cubic equation, where the highest power is three, I should get three solutions.
But what about the other two solutions we expect from a cubic?
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