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Complex numbers are a type of number that expand upon the traditional idea of numbers we use in everyday counting. They are designed to solve equations that do not have real number solutions. A complex number is composed of two parts: a real part and an imaginary part. The standard form for expressing a complex number is written as \( a + bi \), where \( a \) represents the real part and \( b \) represents the imaginary part, with \( i \) being the imaginary unit, satisfying the equation \( i^2 = -1 \).

For instance, in the problem \((7 - 3i) - (6 - i)\), \(7\) and \(6\) are real numbers while \(-3i\) and \(-i\) are imaginary numbers. The beauty of the standard form lies in its simplicity and the ease with which we can perform algebraic operations like addition and subtraction, akin to handling two-dimensional vectors.

Delving deeper into the composition of complex numbers, each part holds its own significance. The real part, signified without an 'i', constitutes the 'normal' number part that can be found on the number line. In contrast, the imaginary part contains the 'i' and represents the magnitude linked to the imaginary unit. This separation allows complex numbers to stretch beyond the one-dimensional number line into a two-dimensional plane, called the complex plane.

In our example, from the expression \((7 - 3i)\), the real part is 7 and the imaginary part is \(-3i\). Additionally, taking the subtrahend \((6 - i)\), the real part is 6 and the imaginary part is \(-i\). Noticing the real and imaginary parts enable us to handle complex numbers systematically.

Performing algebraic operations on complex numbers follows rules akin to those of real numbers, with special consideration given to the imaginary unit 'i'. When we subtract complex numbers, as in the given exercise, we subtract their real parts and their imaginary parts separately. This mirrors vector subtraction, where each component is dealt with individually.

Hence, for the subtraction \((7-3i)-(6-i)\), we distil this down into two separate subtractions: one for the real parts, \(7 - 6\), and another for the imaginary parts, \(-3i - (-i)\), which simplifies to \(-3i + i\) because minus a negative becomes a positive. Summing up our results, we assemble them back into the standard form of a complex number, yielding \(1 - 2i\). Understanding these fundamental principles aids in tackling more intricate operations like multiplication and division of complex numbers with confidence.