Numbering rows within groups in a data frame

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Working with a data frame similar to this: 

set.seed(100)  
df <- data.frame(cat = c(rep("aaa", 5), rep("bbb", 5), rep("ccc", 5)), val = runif(15))             
df <- df[order(df$cat, df$val), ]  
df  

   cat        val  
1  aaa 0.05638315  
2  aaa 0.25767250  
3  aaa 0.30776611  
4  aaa 0.46854928  
5  aaa 0.55232243  
6  bbb 0.17026205  
7  bbb 0.37032054  
8  bbb 0.48377074  
9  bbb 0.54655860  
10 bbb 0.81240262  
11 ccc 0.28035384  
12 ccc 0.39848790  
13 ccc 0.62499648  
14 ccc 0.76255108  
15 ccc 0.88216552

I am trying to add a column with numbering within each group. Doing it this way obviously isn’t using the powers of R: 

 df$num <- 1  
 for (i in 2:(length(df[,1]))) {  
   if (df[i,"cat"]==df[(i-1),"cat"]) {  
     df[i,"num"]<-df[i-1,"num"]+1  
     }  
 }  
 df  

   cat        val num  
1  aaa 0.05638315   1  
2  aaa 0.25767250   2  
3  aaa 0.30776611   3  
4  aaa 0.46854928   4  
5  aaa 0.55232243   5  
6  bbb 0.17026205   1  
7  bbb 0.37032054   2  
8  bbb 0.48377074   3  
9  bbb 0.54655860   4  
10 bbb 0.81240262   5  
11 ccc 0.28035384   1  
12 ccc 0.39848790   2  
13 ccc 0.62499648   3  
14 ccc 0.76255108   4  
15 ccc 0.88216552   5

What would be a good way to do this?

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