# Does 'i' after value stand for 'index', and if so why does it appear as part of the answer?

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Alexander Eaton on 24 Mar 2018
Commented: Alexander Eaton on 25 Mar 2018
I am trying to calculate the Euclidean distance between two images vectors of 4096 values that are contained in a file called allData (see pic) with the command:
Dist = sqrt(sum((allData(1) - allData(3) .^ 2)))
the answer that was returned was
0.0000 +19.1794i
does this answer seem correct or is the 'i' returned because im indexing into a matrix and im just getting returned one of the values from that image vector or am i being returned the distance between the two rows?

Roger Stafford on 24 Mar 2018
Edited: Roger Stafford on 24 Mar 2018
The 'i' in your result indicates a complex answer. I assume you are familiar with complex numbers. The 'i' represents the square root of minus one. This is caused by your attempt to take the square root of a negative number. Apparently allData(1) is less than the square of allData(3).
Roger Stafford on 25 Mar 2018
One simple way is to create a 208-by-208 matrix in which to store all the possible row pair distances:
n = size(allData,1);
D = zeros(n);
for i1 = 1:n
for i2 = 1:n
D(i1,i2) = sqrt(sum((allData(i1,:)-allData(i2,:)).^2));
end
end
Another way is to use the Matlab function 'pdist' which you can read about at: https://www.mathworks.com/help/stats/pdist.html
As to 'complex' numbers, there is an entire mathematical discipline built around them, and much of modern science would probably collapse without them. Ordinary numbers that you learned about in grade school, which in mathematics are referred to as "real" numbers, are those that can be visualized as points along a straight line extending infinitely far in both the positive and the negative directions. Complex numbers can be visualized as points in an infinite plane in which the real numbers are those along the "real" line, and points off this line are the complex numbers. A special complex number is designated as "i" and is the square root of minus one, so that i^2 = -1. Any point in the plane can then be represented in the form "a+b*i" where a and b are real numbers and are analogous to x, y Cartesian coordinates. The four basic arithmetic operations of addition, subtraction, multiplication, and division can then be easily defined. For example, the product of a+b*i and c+d*i where a, b, c, and d are real is given by:
(a+b*i)*(c+d*i) = a*c+a*d*i+b*c*i+b*i*d*i =
a*c+a*d*i+b*c*i+b*d*(-1) = (a*c-b*d)+(a*d+b*c)*i
(The Matlab language is built to handle complex numbers, as you have discovered.)
Alexander Eaton on 25 Mar 2018
thank you Roger for all your help