die Mathe
math
Colloquial short form for the school subject or the scientific field of mathematics.
Ich habe morgen eine wichtige Klausur in Mathe.
I have an important math test tomorrow.

math
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math
Colloquial short form for the school subject or the scientific field of mathematics.
Ich habe morgen eine wichtige Klausur in Mathe.
I have an important math test tomorrow.

'Mathe' (math) is the colloquial and very common short form of 'die Mathematik' (mathematics). In formal or academic texts, you should use the long form.
In der Schule war Mathe mein Lieblingsfach.
At school, math was my favorite subject.
In everyday language, 'Mathe' (math) is often combined with verbs like 'haben' (to have, in the sense of having class), 'lernen' (to study), 'üben' (to practice), or 'verstehen' (to understand). 'Eine Prüfung in Mathe' (a math exam) is also a fixed combination.
Wir haben heute Nachmittag zwei Stunden Mathe.
We have two hours of math this afternoon.
When 'Mathe' (math) is referred to as a school subject or university subject, it often appears without an article, especially after prepositions like 'in' or as a direct object.
Sie ist wirklich gut in Mathe.
She is really good at math.
mathematics
'die Mathematik' and 'die Mathe' mean the same subject or field in many situations. The difference is mainly in style. 'Die Mathematik' sounds complete and fits well in formal texts, at university, or when talking about the science. 'Die Mathe' is the short, colloquial form and sounds especially natural at school and in everyday speech. In an official text, you would tend to use 'Mathematik'; in conversation, many people simply say 'Mathe'.

Für die Mathe hat Max gestern zwei Stunden gelernt.
Max studied for math for two hours yesterday.
An der Universität beschäftigt sich Lena mit der Geschichte der Mathematik.
At university, Lena studies the history of mathematics.