定理（ていり、teiri）
- theorem

The Japanese word 定理 (ていり, teiri) is a noun that translates to "theorem" in English. It refers to a statement that has been proven based on axioms and other previously proven theorems. It is a fundamental concept in mathematics and other fields that use logical deduction. A 定理 provides a definitive and reliable truth within a specific system of reasoning.

⚫︎ピタゴラスの定理は、直角三角形の辺の長さを計算するために使われます。 Translation: The Pythagorean theorem is used to calculate the lengths of the sides of a right triangle. ⚫︎幾何学の授業で多くの定理を学びました。 Translation: I learned many theorems in geometry class. ⚫︎この定理の証明は少し複雑です。 Translation: The proof of this theorem is a bit complicated.

You would use 定理 in any situation where you are discussing mathematical or logical statements that have been proven. This includes academic settings, textbooks, and any conversation involving established facts based on proof. It's crucial when discussing subjects involving deductive reasoning, such as mathematics, physics, and computer science. You would use 定理 when referring to a specific 定理 or discussing 定理 in general.

定理 is typically written in kanji as 定理. The reading is ていり in hiragana. There are no other common ways to write this word, and the okurigana (hiragana suffixes) remains consistent.

There aren't direct synonyms for 定理 that carry the same technical weight in Japanese. However, depending on context, you might use: ⚫︎命題 (めいだい, meidai) - proposition, statement (more general than 定理) ⚫︎公式 (こうしき, koushiki) - formula (often used in mathematical calculations, closely associated with some 定理)

Since 定理 refers to a proven statement, an antonym isn't a single word but rather a concept. Possible contrasts might be: ⚫︎仮説 (かせつ, kasetsu) - hypothesis ⚫︎推測 (すいそく, suisoku) - speculation ⚫︎未証明の命題 (みしょうめいのめいだい) - unproven proposition

三角比の定理は、三角関数を理解するために不可欠です。 ・Translation: The theorems of trigonometric ratios are essential for understanding trigonometric functions. ・Explanation: This sentence illustrates the importance of 定理 (theorems) in understanding more complex mathematical concepts like trigonometric functions.

数学の試験で、いくつかの重要な定理を証明する必要がありました。 ・Translation: In the math exam, I had to prove several important theorems. ・Explanation: This example shows the practical application of 定理 in an academic setting where one has to demonstrate their knowledge of the concept.

The word 定理 is a compound word. The first character, 定 (てい, tei), means "fixed," "established," or "certain." The second character, 理 (り, ri), means "reason," "logic," or "principle." Therefore, 定理 literally translates to "fixed principle" or "established principle," accurately reflecting the meaning of a proven mathematical statement.