Learn Farsi: عیار را بررسی کرده، این الماس را بی‌رنگ می‌دانند. - Having checked the carat, they consider this diamond colorless.

This is a predicative complement, which is the focus of this lesson. With verbs of consideration like dânestan, you simply place the direct object and its marker (in almâs râ) directly before the adjective or bare noun that describes it (bi-rang). No equivalent of 'to be' is needed. The structure is just '[object] + râ + [complement] + [verb]'.

I originally learned that dânestan means 'to know'. How do I know when it translates to 'to consider'?

The meaning shifts based on the sentence structure. When dânestan is followed by a 'that' clause (using ke), it usually means 'to know'. However, when it takes a direct object followed by an adjective or noun complement (like bi-rang in this sentence), its meaning shifts to 'to consider' or 'to reckon'.

Why is râ placed before bi-rang instead of after it?

The placement of râ is what makes the grammar work! If you said in almâs-e bi-rang râ, you would be linking the words with an ezafe, changing the meaning to just 'the colorless diamond' as a single noun phrase. By putting râ immediately after in almâs, you cap off the direct object, making it clear that the following word (bi-rang) is a separate complement.

How does barrasi karde translate to 'having checked'? Isn't that just a past participle?