OK, complex numbers. These are numbers of the form x+i*y where x and y are real numbers and i*i=-1.
ALL the usual rules of algebra still apply, so
(2+3*i)(4-5*i)=2*4 -2*5*i +3*i*4 -3*5*i*i =8 -10*i +12*i -15*(-1) =23+2*i.
Now, if we want (and we should), we can identify the complex number x+i*y with the point of the plane (x,y). In this scheme, addition becomes ordinary vector addition, but multiplication turns out to *add* angles and multiply lengths of the vectors.
So, for example, multiplication by i corresponds to a 90 degree rotation. Since two such multiplications flip the direction of a vector, we get i*i=-1.
The complex numbers have a variety of uses. They simplify notation for many phenomena involving wave motion and are used in electronics extensively because of this. Quantum mechanics has wave functions that are complex valued and this is essential to explain some of the strange behavior of the quantum world.
We can consider the Gaussian integers, which are numbers of the form m+n*i where both m and n are integers. It turns out that many of the classical properties of the integers still hold for Gaussian integers, including things like unique factorization into primes. These turn out to be useful even for questions about ordinary integers and many of their early uses were for that purpose.
ALL the usual rules of algebra still apply, so
(2+3*i)(4-5*i)=2*4 -2*5*i +3*i*4 -3*5*i*i =8 -10*i +12*i -15*(-1) =23+2*i.
Now, if we want (and we should), we can identify the complex number x+i*y with the point of the plane (x,y). In this scheme, addition becomes ordinary vector addition, but multiplication turns out to *add* angles and multiply lengths of the vectors.
So, for example, multiplication by i corresponds to a 90 degree rotation. Since two such multiplications flip the direction of a vector, we get i*i=-1.
The complex numbers have a variety of uses. They simplify notation for many phenomena involving wave motion and are used in electronics extensively because of this. Quantum mechanics has wave functions that are complex valued and this is essential to explain some of the strange behavior of the quantum world.
We can consider the Gaussian integers, which are numbers of the form m+n*i where both m and n are integers. It turns out that many of the classical properties of the integers still hold for Gaussian integers, including things like unique factorization into primes. These turn out to be useful even for questions about ordinary integers and many of their early uses were for that purpose.
