Complex Numbers

Time limit0.5sMemory limit512 MB

Summary

Given two complex numbers written as (b + c*sqrt(d))/a with a shared squarefree d, print their sum, difference, product, and quotient in the same reduced canonical form.

Level: Medium7 of 10
Topics: Math, Number theory, Implementation, Simulation
Solved: No attempts yet

Problem

$\displaystyle{{b_0 + c_0 \sqrt{d_0} \over a_0} + {b_1 + c_1 \sqrt{d_1} \over a_1} i}$

$i = \sqrt{-1}$
For $j=0, \, 1$ :
- $a_j, \, b_j, \, c_j, \, d_j$ are integers
- $a_j > 0$ ; $0 \le d_j \ne 1$
- $\gcd (a_j, \, b_j, \, c_j) = 1$
- ' $c_j = 0$ ' if and only if ' $d_j = 0$ '
- When $d_j > 0$ , $d_j$ has no square divisor greater than $1$
$d_0=d_1$

Given complex numbers $A$ and $B$ written in the form above, write a program that prints the values of $A+B$ , $A-B$ , $A \times B$ , and $A \div B$ .

Input

The first line gives the values $a_0, \, b_0, \, c_0, \, d_0, \, a_1, \, b_1, \, c_1, \, d_1$ of $A$ , one per position.

The second line gives the values $a_0, \, b_0, \, c_0, \, d_0, \, a_1, \, b_1, \, c_1, \, d_1$ of $B$ in the same way.

Output

Print four lines. Line one through line four hold the values $a_0, \, b_0, \, c_0, \, d_0, \, a_1, \, b_1, \, c_1, \, d_1$ for $A+B$ , $A-B$ , $A \times B$ , and $A \div B$ , each expressed in the form described in the problem statement.

Constraints

$B \ne 0$
$A$ and $B$ have the same value of $d_0$
The input satisfies the conditions of the form described in the problem statement
Every number in the input has absolute value at most $10$

Examples1

Example 1

Input

2 5 -3 2 3 6 2 2
6 -4 1 2 2 2 -9 2

Expected output

6 11 -8 2 6 18 -23 2
6 19 -10 2 6 6 31 2
12 22 117 2 36 536 -463 2
258212 -62625 -56274 2 258212 -78659 60431 2