https://www.khanacademy.org/math/class-9-tg/x06d55bfa213a79fd:proofs-in-mathematics/x06d55bfa213a79fd:mathematical-statements/v/different-forms-of-theoretical-statements 2026-02-18T21:20:35.275198076Z Different forms of theoretical statements In this video, we dive into the world of theoretical statements—the backbone of mathematical reasoning. We begin by introducing what a mathematical statement is: a declarative sentence that is either true or false, such as “If a number is even, then it is divisible by 2.” From there, we explore how to negate a statement, flipping its truth value while preserving logical structure. Next, we unpack the converse of a statement—what happens when we reverse the hypothesis and conclusion—and discuss why the converse isn’t always true, even if the original statement is. Finally, we explore the powerful technique of proof by contradiction, where we assume the opposite of what we want to prove and show that this assumption leads to a logical inconsistency. Through intuitive examples and step-by-step reasoning, this video builds your understanding of how mathematicians construct and challenge ideas using formal logic. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 Introduction to theoretical statements 00:58 Negation of a statement 02:53 Converse of a statement 05:08 Proof by contradiction Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Different forms of theoretical statements In this video, we dive into the world of theoretical statements—the backbone of mathematical reasoning. We begin by introducing what a mathematical statement is: a declarative sentence that is either true or false, such as “If a number is even, then it is divisible by 2.” From there, we explore how to negate a statement, flipping its truth value while preserving logical structure. Next, we unpack the converse of a statement—what happens when we reverse the hypothesis and conclusion—and discuss why the converse isn’t always true, even if the original statement is. Finally, we explore the powerful technique of proof by contradiction, where we assume the opposite of what we want to prove and show that this assumption leads to a logical inconsistency. Through intuitive examples and step-by-step reasoning, this video builds your understanding of how mathematicians construct and challenge ideas using formal logic. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 Introduction to theoretical statements 00:58 Negation of a statement 02:53 Converse of a statement 05:08 Proof by contradiction Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy https://cdn.kastatic.org/ka-youtube-converted/iFtUgNM_zIg.mp4/iFtUgNM_zIg.mp4 362 Mathematical statements https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-deductive-and-inductive-reasoning/v/deductive-reasoning-1 2020-07-26T05:38:46Z Inductive & deductive reasoning Sal discusses the difference between inductive and deductive reasoning by considering a word problem. Sal Khan video https://cdn.kastatic.org/googleusercontent/4GduU-VXIB2X6j7_k4kHmm8pIdtaU5zIm59rweDY7x8xQmUmcNSQuDgJW2oit9qFTrZ3lq4msijmFjzldmDjUHs Inductive & deductive reasoning Sal discusses the difference between inductive and deductive reasoning by considering a word problem. https://cdn.kastatic.org/ka-youtube-converted/GEId0GonOZM.mp4/GEId0GonOZM.mp4 120 Mathematical statements https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-deductive-and-inductive-reasoning/v/deductive-reasoning-3 2020-07-26T05:38:46Z Using deductive reasoning Sal uses deductive reasoning to prove an algebraic identity. Sal Khan video https://cdn.kastatic.org/googleusercontent/F9L9gASbCSkqXhun0u7uqORrP-NydAdGWh3DwFrRMwnfvCIYMmCuktDj9dGKOd0VYQ2PIQ-RpZQ6gLSsXYEsRoop Using deductive reasoning Sal uses deductive reasoning to prove an algebraic identity. https://cdn.kastatic.org/ka-youtube-converted/lYLaaMObgkw.mp4/lYLaaMObgkw.mp4 251 Mathematical statements https://www.khanacademy.org/math/algebra-home/alg-series-and-induction/alg-deductive-and-inductive-reasoning/v/inductive-reasoning-2 2020-07-26T05:38:46Z Inductive reasoning Sal analyzes a solution of a mathematical problem to determine whether it uses inductive reasoning. Sal Khan video https://cdn.kastatic.org/googleusercontent/5YTqQ3TNtcJ8C6YtP_8xNcd3pKmhL9e59mPv0v_uqJ9y8eCMFaIYl3Doa-JMupk20gO88Ctfl5FK0G-7LUPCTR8I Inductive reasoning Sal analyzes a solution of a mathematical problem to determine whether it uses inductive reasoning. https://cdn.kastatic.org/ka-youtube-converted/ZSsBh4Ybbj8.mp4/ZSsBh4Ybbj8.mp4 141 Mathematical statements https://www.khanacademy.org/math/class-9-tg/x06d55bfa213a79fd:the-elements-of-geometry/x06d55bfa213a79fd:axioms-and-postulates/v/conjecture-hypothesis 2025-10-06T11:00:50.235589704Z Conjecture (Hypothesis) What’s the difference between a bold mathematical guess and a rock-solid truth? In this video, we explore the fascinating world of conjectures and theorems—starting with the legendary Goldbach’s Conjecture, which has puzzled mathematicians for centuries.Then, we shift gears and prove other conjectures using Euclid’s timeless axioms and postulates, showing how abstract ideas can be grounded in logical certainty. With clear visuals, real-world analogies, and step-by-step reasoning, this lesson makes deep mathematical thinking accessible and engaging. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 What is conjecture? 01:10 Goldbach's conjecture 01:50 Introduction to theorem 03:18 Examples of conjectures Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy Khan Academy video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Conjecture (Hypothesis) What’s the difference between a bold mathematical guess and a rock-solid truth? In this video, we explore the fascinating world of conjectures and theorems—starting with the legendary Goldbach’s Conjecture, which has puzzled mathematicians for centuries.Then, we shift gears and prove other conjectures using Euclid’s timeless axioms and postulates, showing how abstract ideas can be grounded in logical certainty. With clear visuals, real-world analogies, and step-by-step reasoning, this lesson makes deep mathematical thinking accessible and engaging. Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now!Unit link 00:00 What is conjecture? 01:10 Goldbach's conjecture 01:50 Introduction to theorem 03:18 Examples of conjectures Khan Academy India is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We have videos and exercises that have been translated into multiple Indian languages, and 15 million people around the globe learn on Khan Academy every month. Support Us: https://india.khanacademy.org/donate Created by Khan Academy https://cdn.kastatic.org/ka-youtube-converted/WiUSN3A81bg.mp4/WiUSN3A81bg.mp4 323 Mathematical statements https://www.khanacademy.org/math/class-9-tg/x06d55bfa213a79fd:proofs-in-mathematics/x06d55bfa213a79fd:theorems-conjectures-and-axioms/v/how-sure-are-we 2024-12-03T23:53:40.067739643Z How sure are we? In geometry, science, and in many areas of life, we have vocabulary to describe how sure we are. Conjecture: A statement that we believe is always true because we have seen many examples of it and no counterexamples. Postulate: A statement that we believe is always true because people generally agree it is true, but without proof. Theorem: A statement that we believe is always true because we prove it by a series of other definitions, postulates, and theorems. Sal Khan video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw How sure are we? In geometry, science, and in many areas of life, we have vocabulary to describe how sure we are. Conjecture: A statement that we believe is always true because we have seen many examples of it and no counterexamples. Postulate: A statement that we believe is always true because people generally agree it is true, but without proof. Theorem: A statement that we believe is always true because we prove it by a series of other definitions, postulates, and theorems. https://cdn.kastatic.org/ka-youtube-converted/a9TUKHWY3TA.mp4/a9TUKHWY3TA.mp4 470 Mathematical statements https://www.khanacademy.org/math/revision-tg-math-class-9/x6c9499d6e7016acd:week-3/x6c9499d6e7016acd:mathematical-statements/e/revision-mathematical-statements-tg-class-9 2026-02-18T21:20:35.275198076Z Revision mathematical statements TG class 9 . Khan Academy exercise