Being able to work with formulas lets you easily integrate all the parts of a computation.
Here are the eigenvalues of a matrix of numbers.
Mathematica can still compute the eigenvalues even when symbolic parameters are introduced. The result is in effect a compact representation of the eigenvalues for any value of b.
Mathematica's functions are carefully designed so that output from one can easily be used as input to others.
This takes the formula for the eigenvalues and immediately plots it.
You can solve for the value of b at which the first eigenvalue is zero...
Or find the integral from 0 to c.
This finds the series expansion of the result.
This searches numerically for a root.
Being able to work with formulas is also important in summarizing data.
This generates a table of the first 20 primes.
Fit produces an approximate formula.
This computes the sum of the first 40 primes using the approximate formula.
Here is the exact result.
The puts everything together and makes a plot of the difference between exact and approximate results for sums of up to 50 primes.