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# Why does the function real return expressions containing i ?

Asked by Qingbin on 28 Mar 2013

For the following code, CEre is derived using the real function, but it contains i in the expression.

```clc;clear all;
syms s;
syms  T1 T2 w tau1 tau2 real;
e1=(1-T1*s)/(1+T1*s);
e2=(1-T2*s)/(1+T2*s);
A=[0        1       0       0;
0        0       1       0;
0        0       0       1;
-29.17   -56   -36.7   -10.1];
B1=[-1.55   1       0       0;
-1    -0.3      0       0;
0       0       0.5     0;
-0.7    0       -0.34   -2.6];
B2=[0       0       0       0;
1      1.5      4       0;
0       0       0       0;
-0.33     0       0      -1.1];
B3=[0       0       0       0;
0       0       0       0;
0       0       0       0;
-0.08    -0.7     0       -1];
B4=zeros(4,4); B4(4,3)=-3;
n=size(A,1);
tau3=0.169; tau4=0.26;
cez=numden(simplify(det(w*1i*eye(n)-A-B1*e1-B2*e2-B3*(cos(tau3*w)-1i*sin(tau3*w))-B4*(cos(tau4*w)-1i*sin(tau4*w)))));
% ce=collect(cez,s);
cew=subs(cez,s,w*1i)
CEre=real(cew);
CEim=imag(cew);
```

After commenting off the function collect, the function real gives a result without the unity of complex i.

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Answer by Walter Roberson on 28 Mar 2013

I have not tried the above code, but consider

```syms w
real(w*1i)
```

The result may contain "i" because it is possible that w is complex itself.

In this particular case, the result could also be expressed as imag(w), but there are other cases that are more difficult to resolve, such as

```real((w + 1i)^2)
```

## 1 Comment

Qingbin on 29 Mar 2013
```syms  T1 T2 w tau1 tau2 real
```

In the code above, it is defined that w is real. So

```real(w*1i) = 0
```

In fact, using the code above, all the variables are real.

Answer by Ahmed A. Selman on 28 Mar 2013
Edited by Ahmed A. Selman on 29 Mar 2013

Check the math, e.g.:

```...
TheEq=w*1i*eye(n)-A-B1*e1-B2*e2- B3*(cos(tau3*w)-1i*sin(tau3*w))-B4*cos(tau4*w)-1i*sin(tau4*w);
TheDet=det(TheEq);
TheSimple=simplify(TheDet);
cez=numden(TheSimple);
...etc
```

Then see what do

```   TheSimpleEq=simplify(TheEq);
pretty(TheSimpleEq)```

give.

Why do you use (1i) instead of (i) only? If it is not declared before, then (i or j) alone are assumed a unity of complex in Matlab.

Walter Roberson on 28 Mar 2013

1i is safer than i or j, as then you do not need to worry about i or j having been accidentally assigned.

Ahmed A. Selman on 29 Mar 2013

If someone wrote a code, he shouldn't have to worry about accidents :)

Walter Roberson on 29 Mar 2013

Be safe, program defensively. And make it easier for other people to read.