Learn Hardcore German: Die Zahlen im Diagramm sind schlecht, folglich müssen wir unsere Kosten weiter senken. - The numbers in the diagram are bad; consequently we have to keep reducing our costs.

dative = when something is in a place (no movement)
- im Diagramm = in the diagram (location)
accusative = when something is moving into a place
- ins Diagramm = into the diagram (movement into it)

Here it’s clearly a location (the numbers are in the diagram), so dative is used: (in) dem Diagramm → im Diagramm.
Also, Diagramm is a neuter noun (das Diagramm).

Why is it Zahlen and not Nummern?

German distinguishes:

die Zahl (Plural: Zahlen)
= a numerical value, figure, amount
Used for:
- statistics
- charts, diagrams
- financial figures
  Example: Die Zahlen im Diagramm = the figures in the chart
die Nummer (Plural: Nummern)
= an identifying number (like a label or code)
Used for:
- phone numbers (Telefonnummer)
- house numbers
- seat numbers, order numbers

In a business or statistical context, Zahlen is almost always the right word for figures / numbers shown in a diagram.
Nummern im Diagramm would sound like label numbers (e.g. point 1, 2, 3), not financial or performance figures.

What kind of word is folglich, and why does the verb come right after it (folglich müssen wir...)?

folglich is a conjunctive adverb meaning consequently / therefore.

German main clauses follow the verb-second rule:

The finite verb (here: müssen) must be in second position in the clause.
The first position can be:
- the subject (Wir müssen …), or
- an adverbial / other element (Heute müssen wir …, Deshalb müssen wir …, Folgich müssen wir …)

In the sentence:

First position: folglich
Second position (verb): müssen
Then: wir unsere Kosten weiter senken

So the word order folglich müssen wir is required by the verb-second rule.

If you put the subject first instead, you get:

Wir müssen folglich unsere Kosten weiter senken.

Same meaning, just a different emphasis.

Can I replace folglich with deshalb, daher, darum, or also? Are there differences?

Yes, you can replace folglich with other common connectors:

deshalb – therefore, that’s why
daher – therefore, hence
darum – therefore, for that reason
also – so (in the sense of therefore, not the filler well)

All of these would work:

… sind schlecht, deshalb müssen wir unsere Kosten weiter senken.
… sind schlecht, daher müssen wir …
… sind schlecht, darum müssen wir …
… sind schlecht, also müssen wir …

Differences in feel:

folglich – relatively formal, a bit more written/structured
deshalb / daher / darum – very common and neutral
also – common in speech; can sound less formal

Grammatically, they all behave like folglich: they’re conjunctive adverbs and trigger verb-second word order when they start the clause (deshalb müssen wir, daher müssen wir, etc.).

What is the difference between senken and sinken, and how does that explain unsere Kosten?

senken and sinken form a typical transitive/intransitive pair:

senken (regular, transitive) – to lower something
- Requires an accusative object
- Example: Wir senken die Kosten. = We lower the costs.
sinken (irregular, intransitive) – to go down, to sink (by itself)
- No object; the subject itself moves downward
- Example: Die Kosten sinken. = The costs drop.

In your sentence:

senken is used, so we need a direct object.
unsere Kosten is that object, so it’s in the accusative plural.

Nominative: unsere Kosten sind hoch
Accusative: wir müssen unsere Kosten senken

So unsere Kosten is accusative here because senken acts on it directly.

What exactly does weiter mean in Kosten weiter senken, and why not just say Kosten senken?

weiter here is an adverb meaning further / continue to / even more.

Kosten senken = to lower costs (once, in general)
Kosten weiter senken = to lower costs further, keep lowering or reduce them more

So weiter implies that:

costs have already been lowered before, and
now they must be reduced even more.

You could say only Kosten senken, but it would miss that sense of continuation / additional reduction.

Compare a few nuances:

wir müssen die Kosten senken – we must reduce the costs.
wir müssen die Kosten weiter senken – we must keep reducing them / reduce them even more.
wir müssen die Kosten noch weiter senken – we must reduce costs even further still.

Why is schlecht used to describe Zahlen? Could we also say niedrig, schwach, or something else?

With Zahlen in a business context, schlecht expresses an evaluation: the figures are bad / poor.

So:

Die Zahlen sind schlecht.
= The figures are poor (e.g. profit is low, losses are high, growth is weak).

niedrig means low in a more neutral sense:

Die Kosten sind niedrig. – The costs are low (usually good).
Die Preise sind niedrig. – The prices are low.

Using niedrig with Zahlen is possible but it doesn’t automatically carry a positive or negative evaluation; it just describes the level.

Other options:

Die Zahlen sind schwach. – The figures are weak (common in business).
Die Zahlen fallen enttäuschend aus. – The figures are disappointing.
Die Zahlen sehen nicht gut aus. – The figures don’t look good.

In the original sentence, schlecht is short, clear, and directly evaluative, which fits the idea that the bad numbers force cost-cutting.

Why is it die Zahlen and not just Zahlen without an article?

The definite article die is used because both speaker and listener know which numbers are meant: the specific ones in that particular diagram.

die Zahlen im Diagramm
= the specific figures in this diagram (definite, known set)

If you dropped the article and said just Zahlen im Diagramm sind schlecht, it would sound incomplete and unnatural in standard German; it would be like saying numbers in the diagram are bad without specifying the numbers.

German generally uses the definite article more than English when talking about concrete, specific things that are visible or contextually clear.

Why is Kosten plural, and how does that influence unsere Kosten?

Kosten is a plural-only noun in German (a so‑called Pluraletantum):

You normally do not use a singular form like die Kost to mean financial cost.
Standard is: die Kosten = the costs, the expenses.

As a plural noun:

Nominative plural: die Kosten sind hoch
Accusative plural: wir senken die Kosten
With a possessive: unsere Kosten, ihre Kosten, etc.

In your sentence, unsere Kosten is:

plural, because Kosten is plural-only
accusative, because it’s the direct object of senken

So unsere Kosten = our costs/expenses as a plural object.

Why are Zahlen, Diagramm, and Kosten capitalized?

In German, all nouns are capitalized, regardless of their position in the sentence.

In this sentence:

Zahlen – noun (die Zahl, plural)
Diagramm – noun (das Diagramm)
Kosten – noun (plural-only, die Kosten)

So they must all be written with a capital initial letter.

By contrast, weiter, schlecht, and folglich are not nouns (they are adverb/adjective/conjunctive adverb), so they remain lowercase.