Category: Math Dictionary

Казіргі таңда бәсекелестік артықшылықтар әрбір кәсіпкер үшін маңызды. Betandreas платформасы осы жарыста ерекше орын алуда. Оның алға қойған мақсаттары мен клиенттерге ұсынатын артықшылықтары өзге компаниялармен салыстырғанда анық көрінеді.

Betandreas – бұл тек спорттық ставкалармен шектелмейтін, сонымен қатар алуан түрлі ойындар мен мүмкіндіктерді қамтитын жүйе. Осының арқасында ойыншыларға көптеген ұсыныстар мен жеңілдіктер алуға жол ашылады. Бұл артықшылықтар платформаның тартымдылығын арттырады және клиенттерді тартуға көмектеседі.

Компания түрлендіру, жаңарту және клиенттермен қатынасты нығайту бойынша тұрақты жұмыс жүргізеді. Betandreas-тегі әрбір элемент ойын тәжірибесін жақсартуға бағытталған, бұл оны бәсекелестерінен ерекшелендіреді. Ойыншылардың қанағаттанушылығы мен сенімі – платформаның басты артықшылықтары.

Betandreas-тің пайдаланушыларға ұсынатын артықшылықтары

Betandreas-тің пайдаланушыларға артықшылықтары бәсекелестік артықшылықтарын қалыптастыруда маңызды рөл атқарады. Бұл платформа пайдаланушыларға қолайлы интерфейс пен тиімді қызмет көрсетуі арқылы ерекшеленеді. Пайдаланушыларға ыңғайлы навигация жүйесі, ақпараттың жылдамдығы мен дәлдігі Betandreas-тің басты ерекшеліктері болып табылады.

Сондай-ақ, платформа өзінің бәсекелестік артықшылықтары ретінде кең ауқымды ойын түрлерін және жоғары коэффициенттерді ұсынады. Бұл пайдаланушыларға таңдаудың молдығын қамтамасыз етеді, сонымен қатар тиімді ставка жасау мүмкіндігін арттырады.

Betandreas-тің пайдаланушыларға ұсынатын тағы бір маңызды артықшылығы – бұл клиенттермен жұмыс жасау қабілеті. Тұтынушылардың сұраныстарына жедел жауап беру, проблемаларды шешу жылдамдығы платформаның тиімділігін арттырады. Осылайша, Betandreas пайдаланушыларға сенімді және тиімді қызмет көрсетеді.

Ақпараттық қауіпсіздік пен жеке деректерді қорғау да Betandreas-тің бәсекелестік артықшылықтарының бірі болып табылады. Заманауи шифрлау технологиялары мен қауіпсіздік протоколдары пайдаланушылардың ақпараттарының сақталуын қамтамасыз етеді. Бұл Betandreas-ке деген сенімділікті арттырады.

Сонымен қатар, Betandreas пайдаланушыларға арнайы акциялар мен бонустар ұсына отырып, олардың қызығушылығын оятады. Сыйақылар мен арнайы ұсыныстар пайдаланушыларға қосымша артықшылықтар беріп, платформаның тартымдылығын арттырады.

Сатылы бонустар жүйесі: ойыншылар үшін пайда

Betandreas ойыншыларына бірегей сатылы бонустар жүйесін ұсынады, бұл бәсекелестік артықшылықтардың бірі болып табылады. Бұл жүйе ойыншылардың таңдауларын кеңейтеді, әрі оларды платформада белсенді болуға ынталандырады. Әрбір ойыншы өз деңгейіне байланысты түрлі бонустар мен сыйлықтарға ие болады, бұл мейлі жаңа қатысушы, мейлі тәжірибелі ойыншы болсын, барлығына пайдалы.

Сатылы бонустар жүйесінің арқасында, пайдаланушылардың платформамен қарым-қатынасы нығая түседі. Әрбір ойыншы жасаған депозиттер мен белсенділіктеріне қарай сыйақы алады. Бұл тек қана ойыншылардың қаржылық жағынан ұтуына емес, сонымен бірге олардың тәжірибесін жақсартуға мүмкіндік береді.

Жүйе барлық ойыншылар үшін мүмкіндік туғызады, себебі бонустардың әртүрлі деңгейлері болғандықтан, пайдаланушылар өзіне ыңғайлы форматты таңдай алады. Betandreas-тің ұсыныстары әр ойыншыға бәсекелестік артықшылықтарын қамтамасыз етеді, сол арқылы тіпті жаңа жоспарлар мен стратегияларды құруға жәрдемдеседі. Толығырақ ақпаратты https://betandreas-kz.online/ сайтынан алуға болады.

Технологиялық инновациялар: бірегей жүйелер мен қауіпсіздік шаралары

Betandreas пайдаланушыларына бәсекелестік артықшылықтарды ұсынатын бірегей технологиялық шешімдер арқылы ерекшеленеді. Жүйелердің тиімділігі мен қауіпсіздігі – бұл платформадағы ойыншылар үшін басты назарда тұрған аспектілердің бірі.

Платформада қолданылатын бірегей технологиялар мыналарды қамтиды:

• Шифрлау әдістері. Betandreas өзінің пайдаланушыларының деректерін қорғау үшін заманауи шифрлау технологияларын қолданады.
• Тіркелуге арналған биометриялық жүйелер. Пайдаланушылардың қауіпсіздігін арттыру үшін бет-әлпет тану немесе саусақ ізі арқылы тексеру механизмдері енгізілген.
• Жасанды интеллект. Ойын тәжірибесін жақсарту үшін AI алгоритмдері пайдаланылады, бұл ойыншылардың таңдауларын көзге елестетеді.

Қауіпсіздік шаралары Betandreas платформасында ерекше маңызға ие:

1. Халықаралық қауіпсіздік стандарттарына сәйкестік. Платформа барлық қажетті сертификаттарға ие, яғни пайдаланушының сенімділігі жоғары.
2. Үздіксіз мониторинг. Betandreas-те жүйелердің жұмысын бақылайтын арнайы топтар бар, олар ақаулар мен қауіп-қатерлерді тез анықтайды.
3. Техникалық қолдау. Пайдаланушылар кез келген сұрақтары бойынша 24/7 режимінде көмек алуға мүмкіндік береді.

Осы технологиялық инновациялар Betandreas-тің бәсекелестік артықшылықтарын қалыптастырады. Платформа ойыншылар үшін қауіпсіз, сенімді және қолайлы тәжірибені қамтамасыз етеді, осылайша оның жағымды имиджін жоғарылатады.

Клиенттермен байланыс: қолдау қызметінің сапасы мен жылдамдығы

Betandreas компаниясының бәсекелестік артықшылықтарының бірі – клиенттермен байланыс және қолдау қызметінің сапасы. Пайдаланушылар үшін қолжетімді және еркін байланыс жүйесі клиенттердің қажеттіліктерін жылдам анықтауға мүмкіндік береді. Мәселелер туындаған жағдайда, оларды шешуге арналған тәжірибелі мамандар әрдайым дайын.

Сапалы қызмет көрсетудің тағы бір ерекшелігі – жауап беру жылдамдығы. Betandreas қызмет көрсету процесін оңтайландырып, әр клиенттің сұрауына жедел жауап беруді мақсат етеді. Мұндай тез әрі тиімді әрекет ету клиенттердің таңдауын оңайлатады, себебі олар өздерін бағаланған сезінеді.

Клиенттермен байланыс жүйесі, соның ішінде телефон, электрондық пошта және чат арқылы қызмет көрсету, бетандреастың бәсекелестік артықшылықтарының бірі болып табылады. Сервер жүктемесі мен техникалық қиындықтарға қарамастан, Betandreas компаниясы клиенттерге қолдау көрсетуді үздіксіз жүзеге асырады. Бұл таңдаулар клиенттердің сенімін арттырады және олардың бейілділігін күшейтеді.

Қорытындылай келе, Betandreas-тің клиенттермен байланыс жүйесіндегі қолдау қызметінің сапасы мен жылдамдығы, компанияның бәсекелестік артықшылығын көрсететін басты факторлар болып табылады. Пайдаланушылар өздерінің сұраныстарына жедел жауап алатындығын біліп, сенімді түрде ойын ойнауға мүмкіндік алады.

Сұрақ-жауап:

Betandreas-тің бәсекелестік артықшылықтары неде?

Betandreas-тің бәсекелестік артықшылықтары бірнеше аспектілерден тұрады. Біріншіден, ол жедел қызмет көрсету мен клиенттерге бағытталған ұтымды интерфейсті ұсынады. Екіншіден, Betandreas-тің кең ауқымды букмекерлік және казино ойындары бар, бұл клиенттерге түрлі ойын варианттарын ұсынады. Сонымен қатар, Betandreas қауіпсіздік пен сенімділік мәселелеріне ерекше көңіл бөледі, бұл оның клиенттерінің сенімін арттырады.

Betandreas-тің қызметтерін пайдалану үшін қандай шарттар орындалуы қажет?

Betandreas-тің қызметтерін пайдалану үшін ең алдымен, пайдаланушы ретінде тіркелу қажет. Тіркелу барысында жеке деректерді енгізу және жүйенің ережелерімен келісу талап етіледі. Сонымен қатар, пайдаланушының жасы 18-ден жоғары болуы керек. Тіркелгеннен кейін депозитті енгізу арқылы ойынға қатысуға болады. Betandreas түрлі төлем әдістерін қабылдайды, бұл клиенттерге ыңғайлы.

Betandreas-те клиенттер үшін қандай науқандар мен бонустар бар?

Betandreas клиенттеріне арнайы науқандар мен бонустар ұсынады. Жаңадан келгендер үшін приветственный бонус түрінде қосымша қаражат ұсынылады. Сондай-ақ, тұрақты ойыншылар үшін депозит бонусы, кэшбэк және фриспиндер сынды ұсыныстар бар. Бұл науқандар клиенттердің қатысу ынтасын арттырады және Betandreas клиенттеріне жаңа мүмкіндіктер ұсынады.

Betandreas-тің техникасы мен қауіпсіздік шаралары қалай жұмыс істейді?

Betandreas ақпараттық қауіпсіздікке үлкен көңіл бөледі. Платформада SSL шифрлау технологиясы қолданылады, бұл пайдаланушылардың деректерін қорғауға мүмкіндік береді. Сонымен қатар, сайтта төлемдер мен транзакциялар қауіпсіздігіне қатысты қатаң талаптар қойылады. Betandreas-тің лицензиялы болуы және тәуелсіз аудиттерден өтуі де қосымша сенімділік факторы болып табылады.

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Korzystanie z sfinksy.pl powinno być formą rozrywki, a nie sposobem na rozwiązanie problemów finansowych. Użytkownicy są zachęcani do refleksji nad swoją grą i do poszukiwania wsparcia, jeśli zaczynają odczuwać dyskomfort związany z obstawianiem. Tylko w ten sposób możliwe jest delektowanie się grami w bezpieczny i odpowiedzialny sposób.

In een tijd waarin ecologie steeds meer op de voorgrond treedt, is de manier waarop we producten verpakken van groot belang. De uitdaging ligt in het vinden van oplossingen die niet alleen functioneel zijn, maar ook bijdragen aan een betere wereld. Duurzame verpakkingen kunnen een significante rol spelen in het verminderen van afval en het bevorderen van een circulaire economie.

Recycling is een cruciaal element binnen dit proces. Het biedt kansen om materialen hergebruik te maken, wat leidt tot minder belasting van de natuur. Innovatieve verpakkingsmethoden kunnen de levenscyclus van producten verlengen en tegelijkertijd de impact op het milieu minimaliseren. Het is tijd om na te denken over creatieve en praktische manieren om onze verpakkingen te herzien en te transformeren voor een groenere toekomst.

Biologisch afbreekbare materialen: voordelen en toepassingen

Biologisch afbreekbare materialen zijn een innovatieve oplossing voor milieuproblemen die voortkomen uit traditionele verpakkingen. Deze materialen zijn ontworpen om, na gebruik, op natuurlijke wijze af te breken, wat zorgt voor een vermindering van afval en een positieve impact op de ecologie.

Een van de belangrijkste voordelen van biologisch afbreekbare materialen is hun vermogen om de behoefte aan recycling te verminderen. Door gebruik te maken van materialen die op korte termijn afbreken, wordt de druk op recyclinginfrastructuren verlicht. Dit helpt niet alleen het milieu te beschermen, maar vermindert ook de kosten en tijd die gemoeid zijn met het recyclen van traditionele verpakkingen.

Toepassingen van biologisch afbreekbare materialen zijn divers. Ze kunnen worden gebruikt in voedselverpakkingen, tassen en zelfs in bouwmaterialen. Het gebruik van deze materialen in consumentenproductverpakkingen draagt bij aan een schonere toekomst en bevordert een ecologische levensstijl.

Door te kiezen voor biologisch afbreekbare materialen, worden bedrijven en consumenten aangemoedigd om bewustere keuzes te maken. Dit bevordert een circulaire economie, waarin producten niet alleen efficiënt worden vervaardigd, maar ook duurzaam zijn gedurende hun levenscyclus. Meer informatie over duurzame initiatieven vind je op https://mylifeherbalife.com/.

Innovaties in herbruikbare verpakkingen voor consumenten

De opkomst van herbruikbare verpakkingen biedt een waardevolle oplossing voor de problemen rondom plasticafval. Duurzame alternatieven zijn essentieel voor een betere ecologie en het verminderen van de hoeveelheid afval die ons milieu schaadt. Innovaties in deze sector richten zich op gebruiksvriendelijkheid en functionaliteit, zodat consumenten gestimuleerd worden om herbruikbare opties te kiezen.

Een voorbeeld van zo’n innovatie is het ontwerp van slimme herbruikbare verpakkingen die meerdere keren kunnen worden gebruikt zonder in te boeten op kwaliteit. Deze verpakkingen zijn vaak vervaardigd uit duurzame materialen die eenvoudig gerecycled kunnen worden, wat de cirkel van afvalvermindering sluit. Door gebruik te maken van modulaire systemen kan de verpakking eenvoudig worden aangepast aan verschillende producten.

Daarnaast ontstaan er applicaties die consumenten aanmoedigen om bij lokale winkels hun herbruikbare verpakkingen in te zetten. Deze systemen zorgen ervoor dat consumenten sneller overgaan tot het afstemmen van hun uitgaven op een duurzamer levenspatroon. Door het aanbieden van kortingen voor gebruikers van herbruikbare verpakkingen wordt niet alleen afval verminderd, maar ook de gemeenschap betrokken bij het verbeteren van recycleerbaarheid en bewuste keuzes bij de inkoop.

Innovaties in herbruikbare verpakkingen verminderen niet alleen plasticafval, maar bieden ook een kans voor educatie over recycling. Door consumenten bewust te maken van de impact van hun keuzes, kunnen zij bijdragen aan een betere ecologie en een schonere toekomst. Dergelijke initiatieven zijn stappen richting een meer duurzame samenleving.

Kostenbesparingen door duurzame verpakkingsstrategieën voor bedrijven

Duurzame verpakkingsstrategieën kunnen aanzienlijke kostenbesparingen opleveren voor bedrijven, niet alleen in termen van materiaalgebruik maar ook door efficiëntie in productieprocessen.

Door over te stappen op milieuvriendelijke verpakkingsoplossingen kunnen bedrijven hun afhankelijkheid van traditionele plastic verminderen, wat leidt tot lagere uitgaven voor grondstoffen. Daarnaast draagt het gebruik van recycleerbare materialen bij aan het verminderen van afvalverwerkingskosten.

• Vermindering van grondstofkosten: Biologisch afbreekbare en recycleerbare materialen zijn vaak goedkoper in aanschaf en verwerking op de lange termijn.
• Efficiëntere productieketens: Duurzame verpakkingen kunnen leiden tot snellere productieprocessen, waardoor arbeids- en energiekosten dalen.
• Verlaagde afvalkosten: Het verminderen van plasticafval door in te zetten op recycling en herbruikbare verpakkingen kan bedrijven geld besparen op afvalbeheer.
• Verbeterde merkwaardering: Klanten zijn steeds meer op zoek naar duurzame producten. Bedrijven die in lijn zijn met deze waarden kunnen hogere verkoopcijfers verwachten, wat de winst verhoogt.

Door een annalyse uit te voeren van hun huidige verpakkingsstrategieën kunnen bedrijven kansen identificeren voor kostenbesparing en tegelijkertijd bijdragen aan een betere milieu-impact. Dit niet alleen ten goede van hun financiële gezondheid, maar ook voor de planeet.

Regelgevingen en trends in duurzame verpakking binnen de EU

De Europese Unie heeft ambitieuze standaarden opgesteld om duurzame verpakkingen te bevorderen. Regelgeving zoals de Europese Richtlijn verpakkingen en verpakkingsafval stelt eisen aan de recycling en het verminderen van afval. Dit zorgt voor een betere omgang met hulpbronnen en ondersteunt de overgang naar een circulaire economie.

Een belangrijke trend binnen de EU is de verplichting voor producenten om gerecycleerde materialen te gebruiken in hun verpakkingen. Deze maatregel vermindert de druk op grondstoffen en stimuleert de ecologie door recyclingprocessen te optimaliseren. Daarnaast worden nieuwe verpakkingsmaterialen ontwikkeld die specifiek gericht zijn op recycleerbaarheid en milieu-impact.

Verder zijn er initiatieven waarbij consumenten en bedrijven worden aangemoedigd om te kiezen voor duurzame alternatieven. Dit gebeurt door middel van voorlichting en incentives. De aandacht voor milieu, ecologische impact en duurzaamheid neemt toe, wat zichtbaar is in de innovaties die bedrijven toepassen in hun verpakkingsstrategieën.

De EU komt ook met richtlijnen over het verbod op bepaalde wegwerpplastic verpakkingen, wat een directe aanzet is tot het gebruik van alternatieven. Deze regels stimuleren het gebruik van herbruikbare en biologisch afbreekbare materialen, wat bijdraagt aan een verminderde milieu-impact van de verpakkingssector.

Door deze regelgeving en trends wordt de weg vrijgemaakt voor de ontwikkeling van duurzame verpakkingsoplossingen die niet alleen aan de behoeften van de consument voldoen, maar ook bijdragen aan een betere toekomst voor onze planeet.

El proceso de selección en el ámbito educativo es una etapa crucial que determina la calidad de los profesionales que ingresan al sistema. Cada vez más instituciones reconocen la importancia de establecer criterios claros y justos para evaluar a los aspirantes, asegurando así que se cumplan los estándares necesarios.

En este contexto, la ugt educación juega un papel fundamental, no solo en la creación de normativas, sino también en la formación de los educadores en prácticas competitivas. Comprender cómo funcionan los concursos de méritos es esencial para quienes buscan mejorar sus oportunidades dentro de este sector.

A través de esta guía, se explorarán los aspectos más relevantes de los concursos de méritos, brindando herramientas y recomendaciones que ayudarán a los educadores a prepararse adecuadamente para este importante proceso. La transparencia y la equidad son valores que deben primar en cada etapa de selección, y esta guía se propone contribuir a su fortalecimiento.

Requisitos y criterios de evaluación en los concursos de méritos

Los concursos de méritos en la educación requieren que los aspirantes cumplan con ciertos requisitos previos, que pueden incluir niveles educativos específicos, experiencia laboral relevante y la provisión de documentación que respalde sus logros. Esta documentación puede abarcar títulos académicos, certificados de formación continua y pruebas de experiencia en el ámbito educativo.

La evaluación de méritos se lleva a cabo mediante criterios claramente definidos que permiten seleccionar a los candidatos más idóneos para las oportunidades laborales disponibles. Estos criterios pueden incluir la formación académica, la experiencia profesional, así como los antecedentes de trabajos anteriores en el sector educativo. Las entidades encargadas de estos concursos se aseguraran de que el proceso sea transparente y justo, evaluando a todos los candidatos de acuerdo con los mismos estándares.

Es fundamental que los aspirantes se informen sobre los requisitos específicos de cada concurso, así como de los criterios de evaluación, que pueden variar dependiendo de la institución y del puesto ofertado. Para más información sobre estas oportunidades laborales y los procedimientos asociados, se puede visitar https://feteugtcantabria.es/.

Documentación necesaria para participar en los concursos educativos

Para participar en los concursos de méritos en la educación, es fundamental contar con la documentación adecuada que respalde la candidatura. Cada convocatoria puede tener especificaciones particulares, pero hay elementos comunes que generalmente se requieren.

Uno de los principales requisitos de participación es la presentación de una solicitud formal, que puede incluir la declaración de los méritos y la intención de participar. Además, es necesario anexar documentos que acrediten la formación académica, como títulos y certificados de estudios.

La experiencia laboral también tiene un papel importante en el proceso de selección. Los aspirantes deben presentar comprobantes de su trayectoria profesional, como contratos, cartas de recomendación o certificados de trabajo, que validen los años de servicio en el ámbito educativo.

Asimismo, se suele solicitar un currículum vitae actualizado que detalle las habilidades y logros alcanzados a lo largo de la carrera profesional. Este documento debe ser claro y conciso, resaltando los aspectos más relevantes para el concurso.

En algunos casos, se pueden requerir certificados de formación continua o especializaciones que refuercen la candidatura. Estos documentos ayudan a demostrar el compromiso del participante con su desarrollo profesional y contribuyen a valorar su idoneidad.

Finalmente, es importante revisar las bases de cada convocatoria, ya que la documentación solicitada puede variar. Asegurarse de presentar todos los documentos requeridos en el formato adecuado es clave para maximizar las posibilidades de éxito en el proceso de selección.

Estrategias para mejorar la puntuación en concursos de méritos

Participar en concursos de méritos requiere una preparación sólida y estratégica. Una de las primeras acciones es conocer a fondo el proceso de selección y los criterios específicos de evaluación de méritos. Esto permite enfocar los esfuerzos en áreas clave que impactarán directamente en la puntuación.

La mejora profesional continua es fundamental. Asistir a cursos, talleres y actividades de formación que estén alineados con los requisitos del concurso puede sumar puntos significativos. Documentar estas experiencias y su impacto en la práctica docente es igualmente importante.

El desarrollo de un portafolio que muestre logros, proyectos ejecutados y innovaciones aplicadas en el ámbito educativo puede constituir una herramienta poderosa. Este portafolio debe estar bien organizado y presentar evidencias claras de cómo cada actividad contribuye al avance en el campo educativo.

Además, es recomendable practicar el manejo del tiempo durante las evaluaciones o entrevistas, asegurándose de comunicar de manera clara y concisa las competencias y experiencias relevantes. La preparación de simulaciones puede resultar útil para fortalecer la confianza en el desempeño.

Por último, contar con el apoyo de colegas o mentores puede ofrecer perspectivas valiosas. Compartir experiencias y recibir retroalimentación sobre la presentación de la propia trayectoria puede contribuir a mejorar la propuesta general presentada durante los concursos de méritos.

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Zermelo-Fraenkel set theory: the standard form of set theory and the most common foundation of modern mathematics, based on a list of nine axioms (usually modified by a tenth, the axiom of choice) about what kinds of sets exist, commonly abbreviated together as ZFC

Zeta function: A function based on an infinite series of reciprocals of exponents (Riemann’s zeta function is the extension of Euler’s simple zeta function into the domain of complex numbers)

Venn diagram: a diagram where sets are represented as simple geometric figures (often circles), and overlapping and similar sets are represented by intersections and unions of the figures

Vector space: a three-dimensional area where vectors can be plotted, or a mathematical structure formed by a collection of vectors

Vector: a physical quantity having magnitude and direction, represented by a directed arrow indicating its orientation in space

Type theory: an alternative to naive set theory in which all mathematical entities are assigned to a type within a hierarchy of types, so that objects of a given type are built exclusively from objects of preceding types lower in the hierarchy, thus preventing loops and paradoxes

Trinomial: an algebraic equation with 3 terms, e.g. 3x + 5y + 8z; 3x³ + 2x² + x; etc

Trigonometry: the branch of mathematics that studies the relationships between the sides and the angles of right triangles, and deals with and with the trigonometric functions (sine, cosine, tangent and their reciprocals)

Triangular number: a number which can be represented as an equilateral triangle of dots, and is the sum of all the consecutive numbers up to its largest prime factor – it can also be calculated as ^{n(n + 1)}⁄₂, e.g. 15 = 1 + 2 + 3 + 4 + 5 = ^{5(5 + 1)}⁄₂

Transfinite numbers: cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite

Transcendental number: an irrational number that is “not algebraic”, i.e. no finite sequence of algebraic operations on integers (such as powers, roots, sums, etc.) can be equal to its value, examples being π and e. For example, √2 is irrational but not transcendental because it is the solution to the polynomial x² = 2.

Topology: the field of mathematics concerned with spatial properties that are preserved under continuous deformations of objects (such as stretching, bending and morphing, but not tearing or gluing)

Term: in an algebraic expression or equation, either a single number or variable, or the product of several numbers and variables separated from another term by a + or – sign, e.g. in the expression 3 + 4x + 5yzw, the 3, the 4x and the 5yzw are all separate terms

Theorem: a mathematical statement or hypothesis which has been proved on the basis of previously established theorems and previously accepted axioms, effectively the proof of the truth of a statement or expression

Tensor: a collection of numbers at every point in space which describe how much the space is curved, e.g. in four spatial dimensions, a collection of ten numbers is needed at each point to describe the properties of the mathematical space or manifold, no matter how distorted it may be

Symmetry: the correspondence in size, form or arrangement of parts on a plane or line (line symmetry is where each point on one side of a line has a corresponding point on the opposite side, e.g. a picture a butterfly with wings that are identical on either side; plane symmetry refers to similar figures being repeated at different but regular locations on the plane)

Subset: a subsidiary collection of objects that all belong to, or is contained in, an original given set, e.g. subsets of {a, b} could include: {a}, {b}, {a, b} and {}

Surd: the n-th root a number, such as √5, the cube root of 7, etc

Spherical geometry: a type of non-Euclidean (elliptic) geometry using the two-dimensional surface of a sphere, where a curved geodesic (not a straight line) is the shortest paths between points

Spherical trigonometry: a branch of spherical geometry which deals with polygons (especially triangles) on the sphere, and the relationships between their sides and angles

Simultaneous equations: a set or system of equations containing multiple variables which has a solution that simultaneously satisfies all of the equations (e.g. the set of simultaneous linear equations 2x + y = 8 and x + y = 6, has a solution x = 2 and y = 4)

Slope: the steepness or incline of a line, determined by reference to two points on the line, e.g. the slope of the line y = mx + b is m, and represents the rate at which y is changing per unit of change in x

Significant digits: the number of digits to consider when using measuring numbers, those digits that carry meaning contributing to its precision (i.e. ignoring leading and trailing zeros)

Set: a collection of distinct objects or numbers, without regard to their order, considered as an object in its own right

Sequence: an ordered set whose elements are usually determined based on some function of the counting numbers, e.g. a geometric sequence is a set where each element is a multiple of the previous element; an arithmetic sequence is a set where each element is the previous element plus or minus a number

Self-similarity: object is exactly or approximately similar to a part of itself (in fractals, the shapes of lines at different iterations look like smaller versions of earlier shapes)

Reciprocal: a number which, when multiplied by x yields the multiplicative identity 1, and can therefore be thought of as the inverse of multiplication, e.g. the reciprocal of x is ¹⁄_x, the reciprocal of ³⁄₅ is ⁵⁄₃

Riemannian geometry: a non-Euclidean geometry that studies curved surfaces and differentiable manifolds in higher dimensional spaces

Real numbers: all numbers (including natural numbers, integers, decimals, rational numbers and irrational numbers) which do not involve imaginary numbers (multiples of the imaginary unit i, or the square root of -1), may be thought of as all points on an infinitely long number line

Rational numbers: numbers that can be expressed as a fraction (or ratio) ^a⁄_b of two integers (the integers are therefore a subset of the rationals), or alternatively a decimal which terminates after a finite number of digits or begins to repeat a sequence

Quintic equation: a polynomial having a degree of 5 (i.e. the highest power is 5), of the form ax⁵ + bx⁴ + cx³ + dx² + ex + f = 0, not solvable by factorization into radicals for all rational numbers

Quaternions: a number system that extends complex numbers to four dimensions (so that an object is described by a real number and three complex numbers, all mutually perpendicular to each other), which can be used to represent a three-dimensional rotation by just an angle and a vector

Quartic equation: a polynomial having a degree of 4 (i.e. the highest power is 4), of the form ax⁴ + bx³ + cx² + dx + e = 0, the highest order polynomial equation that can be solved by factorization into radicals by a general formula

Quadrature: the act of squaring, or finding a square equal in area to a given figure, or finding the area of a geometrical figure or the area under a curve (such as by a process of numerical integration)

Quadratic equation: a polynomial equation with a degree of 2 (i.e. the highest power is 2), of the form ax² + bx + c = 0, which can be solved by various methods including factoring, completing the square, graphing, Newton’s method and the quadratic formula

Pythagorean triples: groups of three positive integers a, b and c such that the a² + b² = c² equation of Pythagoras’ theorem, e.g. ( 3, 4, 5), ( 5, 12, 13), ( 7, 24, 25), ( 8, 15, 17), etc

Pythagoras’ (Pythagorean) theorem: the square of the hypotenuse of a right angled triangle is equal to the sum of the squares of the two sides (a² + b² = c²)

Probability theory: the branch of mathematics concerned with analysis of random variables and events, and with the interpretation of probabilities (the likelihood of an event happening)

Plane: a flat two-dimensional surface (physical or theoretical) with infinite width and length, zero thickness and zero curvature

Prime numbers: integers greater than 1 which are only divisible by themselves and 1

Projective geometry: a kind of non-Euclidean geometry which considers what happens to shapes when they are projected on to a non-parallel plane, e.g. a circle may be projected into an ellipse or a hyperbola

Polar coordinates: a two-dimensional coordinate system in which each point on a plane is determined by its distance r from a fixed point (e.g. the origin) and its angle θ (theta) from a fixed direction (e.g the x axis)

Polynomial: an algebraic expression or equation with more than one term, constructed from variables and constants using only the operations of addition, subtraction, multiplication and non-negative whole-number exponents, e.g. 5x² – 4x + 4y + 7

Platonic solids: the five regular convex polyhedra (symmetrical 3-dimensional shapes): the tetrahedron (made up of 4 regular triangles), the octahedron (made up of 8 triangles), the icosahedron (made up of 20 triangles), the cube (made up of 6 squares) and the dodecahedron (made up of 12 pentagons)

Permutation: a particular ordering of a set of objects, e.g. given the set {1, 2, 3}, there are six permutations: {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, and {3, 2, 1}

Pi (π): the ratio of a circumference of a circle to its diameter, an irrational (and transcendental) number approximately equal to 3.141593…

Periodic function: a function that repeats its values in regular intervals or periods, such as the trigonometric functions of sine, cosine, tangent, etc

Perfect number: a number that is the sum of its divisors (excluding the number itself), e.g. 28 = 1 + 2 + 4 + 7 + 14

Pascal’s Triangle: a geometrical arrangement of the coefficients of the polynomial expansion of a binomial power of the form (x + y)ⁿ as a symmetrical triangle of numbers

Partial differential equation: a relation involving an unknown function with several independent variables and its partial derivatives with respect to those variables

Paradox: a statement that appears to contradict itself, suggesting a solution which is actually impossible

Parabola: a type of conic section curve, any point of which is equally distant from a fixed focus point and a fixed straight line

Ordinal numbers: an extension of the natural numbers (different from integers and from cardinal numbers) used to describe the order type of sets i.e. the order of elements within a set or series

Number theory: the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular

Number line: a line on which all points correspond to real numbers (a simple number line may only mark integers, but in theory all real numbers to +/- infinity can be shown on a number line)

Normal (Gaussian) distribution: a continuous probability distribution in probability theory and statistics that describes data which clusters around the mean in a curved “bell curve”, highest in the middle and quickly tapering off to each side

Non-Euclidean geometry: geometry based on a curved plane, whether elliptic (spherical) or hyperbolic (saddle-shaped), in which there are no parallel lines and the angles of a triangle do not sum to 180°

Non-commutative algebra: an algebra in which a x b does not always equal b x a, such as that used by quaternions

Negative numbers: any integer, ration or real number which is less than 0, e.g. -743, -1.4, -√5 (but not √-1, which is an imaginary or complex number)

Natural numbers: the set of positive integers (regular whole counting numbers), sometimes including zero

Monomial: an algebraic expression consisting of a single term (although that term could be an exponent), e.g. y = 7x, y = 2x³

Modulus: a number by which two given numbers can be divided by integer division, and produce the same remainder, e.g. 38 ÷ 12 = 3 remainder 2, and 26 ÷ 12 = 2 remainder 2, therefore 38 and 26 are congruent modulo 12, or (38 ≡ 26) mod 12

Method of exhaustion: a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape (a precursor to the methods of calculus)

Modular arithmetic: a system of arithmetic for integers, where numbers “wrap around” after they reach a certain value (the modulus), e.g. on a 12-hour clock, 15 o’clock is actually 3 o’clock (15 = 3 mod 12)

Mersenne primes: prime numbers that are one less than a power of 2, e.g. 3 (2² – 1); 7 (2³ – 1); 31 (2⁵ – 1); 127 (2⁷ – 1); 8,191 (2¹³ – 1); etc – many, but not all, Mersenne numbers are primes, e.g. 2,047 = 2¹¹ – 1 = 23 x 89, so 2,047 is a Mersenne number but not a Mersenne prime

Mersenne number: numbers that are one less than 2 to the power of a prime number, e.g. 3 (2² – 1); 7 (2³ – 1); 31 (2⁵ – 1); 127 (2⁷ – 1); 8,191 (2¹³ – 1); etc

Matrix: a rectangular array of numbers, which can be added, subtracted and multiplied, and used to represent linear transformations and vectors, solve equations, etc

Manifold: a topological space or surface which, on a small enough scale, resembles the Euclidean space of a specific dimension (called the dimension of the manifold), e.g. a line and a circle are one-dimensional manifolds; a plane and the surface of a sphere are two-dimensional manifolds; etc

Mandelbrot set: a set of points in the complex plane, the boundary of which forms a fractal, based on all the possible c points and Julia sets of a function of the form z² + c (where c is a complex parameter)

Magic square: a square array of numbers where each row, column and diagonal added up to the same total, known as the magic sum or constant (a semi-magic square is a square numbers where just the rows and columns, but not both diagonals, sum to a constant)

Logicism: the theory that mathematics is just an extension of logic, and that therefore some or all mathematics is reducible to logic

Logic: the study of the formal laws of reasoning (mathematical logic the application of the techniques of formal logic to mathematics and mathematical reasoning, and vice versa)

Logarithm: the inverse operation to exponentiation, the exponent of a power to which a base (usually 10 or e for natural logarithms) must be raised to produce a given number, e.g. because 1,000 = 10³, the log₁₀ 100 = 3

Linear regression: a technique in statistics and probability theory for modelling scattered data by assuming an approximate linear relationship between the dependent and independent variables

Line: in geometry, a one-dimensional figure following a continuous straight path joining two or more points, whether infinite in both directions or just a line segment bounded by two distinct end points

Linear equation: an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable, and whose graph is therefore a straight line, e.g. y = 4, y = 5x + 3

Least squares method: a method of regression analysis used in probability theory and statistics to fit a curve-of-best-fit to observed data by minimizing the sum of the squares of the differences between the observed values and the values provided by the model

Limit: the point towards which a series or function converges, e.g. as x becomes closer and closer to zero, ^(sin x)⁄_x becomes closer and closer to the limit of 1

Knot theory: an area of topology that studies mathematical knots (a knot is a closed curve in space formed by interlacing a piece of “string” and joining the ends)

Julia set: the set of points for a function of the form z² + c (where c is a complex parameter), such that a small perturbation can cause drastic changes in the sequence of iterated function values and iterations will either approach zero, approach infinity or get trapped in loop

Integration: the operation in calculus (inverse to the operation of differentiation) of finding the integral of a function or equation

Irrational numbers: numbers that can not be represented as decimals (because they would contain an infinite number of non-repeating digits) or as fractions of one integer over another, e.g. π, √2, e

Integers: whole numbers, both positive (natural numbers) and negative, including zero

Integral: the area bounded by a graph or curve of a function and the x axis, between two given values of x (definite integral), found by the operation of integration

Infinity: a quantity or set of numbers without bound, limit or end, whether countably infinite like the set of integers, or uncountably infinite like the set of real numbers (represented by the symbol ∞)

Infinitesimal: quantities or objects so small that there is no way to see them or to measure them, so that for all practical purposes they approach zero as a limit (an idea used in the developement of infinitesimal calculus)

Infinite series: the sum of an infinite sequence of numbers (which are usually produced according to a certain rule, formula or algorithm)

Imaginary numbers: numbers in the form bi, where b is a real number and i is the “imaginary unit”, equal to √-1 (i.e. i² = -1)

Identity: an equality that remains true regardless of the values of any variables that appear within it, e.g. for multiplication, the identity is one; for addition, the identity is zero

Hyperbolic geometry: a non-Euclidean geometry based on a saddle-shaped plane, in which there are no parallel lines and the angles of a triangle sum to less than 180°

Hyperbola: a smooth symmetrical curve with two branches produced by the section of a conical surface

Hilbert problems: an influential list of 23 open (unsolved) problems in mathematics described by David Hilbert in 1900

Group theory: the mathematical field that studies the algebraic structures and properties of groups and the mappings between them

Group: a mathematical structure consisting of a set together with an operation that combines any two of its elements to form a third element, e.g. the set of integers and the addition operation form a group

Graph theory: a branch of mathematics focusing on the properties of a variety of graphs (meaning visual representations of data and their relationships, as opposed to graphs of functions on a Cartesian plane)