# 8.5: Summary - Mathematics

8.5: Summary - Mathematics

## Chapter 8 - Personal Finance - 8.5 Annuities, Methods of Saving, and Investments - Exercise Set 8.5 - Page 537: 22

(1+frac)^A$is the final amount in the account$P$is the principal (the amount of money invested)$r$is the interest rate$n$is the number of times per year the interest is compounded$t$is the number of years We can find the total amount in the account at the end of 25 years when we invest a lump sum at a rate of 6.5% compounded annually.$A = P

(1+frac<0.065><1>)^<(1)(25)>A = $193,108$ After 25 years, there will be $$193,108 in the account. This is the formula we use to calculate the value of an annuity: A = frac<> [(1+frac)^-1]>> A is the future value of the annuity P is the amount of the periodic deposit r is the interest rate n is the number of times per year the interest is compounded t is the number of years A = frac<> -1]><1>> A = 94,220 The value of the annuity is$$94,220$We can calculate the difference between the lump-sum investment and the value of the annuity. $$193,108 - 94,220 = 98,888 We will have$$98,888$ more from the lump-sum investment than from the annuity. The lump-sum investment earns 98,888$more in interest than the annuity. ##### Update this answer! You can help us out by revising, improving and updating this answer. After you claim an answer you&rsquoll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. ## 8.5: Summary - Mathematics 1) KNOWLEDGE – is the foundation of all things in existence. It is symbolic to the Asiatic Black man for the face he is the foundation to his family. He is the center of his family controlling them with the knowledge of himself. Knowledge is also symbolic to the Sun for the fact the Sun is the foundation to the Solar System. the Sun is the center of the Solar System controlling all the planets with its gravitational pull. It is broken down into two words, know and Ledge. Once you master the Ledge you mastered knowledge itself. Knowledge means to know the ledge and not to fall of the edge into the pits of ignorance. Ignorance means to ignore the facts. Knowledge is the foundation of all intelligence, all experience and all thought. It allows the righteous black man to stand upright on his square at all times being free of the devils. It is to know all truths and facts about everything in life. Once you deviate from the path of knowledge, you have deviated from the path of Righteousness, for the fact knowledge is as infinite as Allah Himself. 2) WISDOM – is the wise words spoken by the wise person such as I, to show and prove the knowledge of Self and the Existence of God which is myself and the existence of the Father Allah, the Original Mind that everything into existence. Wisdom is symbolic t the Asiatic Black Woman for the fact she is secondary but most necessary for the fact she bring forth the best part of life, which is the Understanding or the Asiatic Black Child. It is also symbolic to the Moon for the she receives her Light from the Sun meaning that her light is secondary. Wisdom is broken down into two words Wise and Dome, Once you wise your dome you can wise the dome of the Blind, deaf and Dumb, to make them see, hear and speak. It is the supreme manifestation of thought which releases the dumb from the confinement of darkness which is the black from the Devil’s Society and way of living. 3) UNDERSTANDING – is the clear picture one draws up in the mind of knowledge and wisdom which is perfectly understood by the all seeing eye, which is the Mind or the Third eye. It is symbolic to the Asiatic Black Child for the fact it takes knowledge and Wisdom to bring forth the Understanding. It is also the Star. the star shines its own light by manifesting the truth by using knowledge and wisdom. It is comprehension (the ability to understand truth or facts) to come to know about things you deal with each and everyday in each and every way. it is the ability to think and learn. It is the interpretation of known facts possessing or showing intelligence. 4) CULTURE or FREEDOM – CULTURE is the way of life that you live. The culture I live is the Divine Culture of Islam. Islam is the way of life that each and every black family should live. Islam broken down in its true nature is I Self Lord And Master or simply I-God the Supreme Being Black man of Asia who is Islam. He is Islam and Islam is him. Islam is also mathematics for the Black woman to live them our. Islam means to the Black Women, I-Sincerely-Love-Allah’s Mathematics, which is the acceptance of a Black Woman into Islam because one must love Islam to live it out. And it must be lived out right and exact or it will not serve its purpose to you because the nation of Islam is all wise and does everything right and exact. No man can come between you and your culture. The doesn’t want the Blackman to know or have his culture. With Islam you want for your brother as you want for yourself. FREEDOM is to free your dome from the Devil’s civilization. It is to leave from a state of mental death and power which is Christianity or any other Religion, into a state of righteousness, love, peace and happiness which is the culture the Divine culture of Islam. Islam guides you from that mental state of unawareness to that mental state of awareness and consciousness. Freedom is the very nature of the divine culture of Islam. The truth of Islam will set you free from all impurities and unrighteousness because the Divine nature of Islam is Righteousness. Freedom is to free the dome of the wickedness which lies within. It is liberty, independence and unrestrictive ness. To be able to choose without being told how, when or what. We were freed from the chains f slavery in the 1800’s, but were mentally, socially and politically still slaves, although it is a step closer to freedom, you are not free until you are Free mentally. 5) POWER or REFINEMENT – POWER is the truth and the truth is the power square, the square of pure righteousness. It is the truth which is the light that brings one out of the triple stages of darkness into their true way of life which is Islam. it gives the wise man the ability to control and refine all that comes within his circumference. The power lies within the Black man mentally, as well as physically, because within his mentality and intelligence, lies supreme powers and forces which created all things in existence through the science of Supreme Mathematics, which all things in existence can be shown and proved through, when I say all things, I mean the Universe because the Universe is everything )all things) The five parts of the black man’s body consist of 72 degrees which shows and proves God’s wisdom is also to refine ones mentality and ways and actions. To be free from all impurities and unrighteousness is refinement. REFINEMENT is to purify yourself both physically and mentally thought the Culture of Islam. it is to be purified in other words Holy, and Holy is something that is not mixed, tampered or diluted with in any form. 6) EQUALITY – Is to be equal in all things in existence such as knowledge, wisdom and understanding. When you share your knowledge through wisdom, and bring forth an understanding It only shows and proves equality. Knowledge (1), Wisdom (2) and Understanding (3) is all being born to equality. Equality means to be equal in everything which is the father’s ability to deal equal with the life nourishing which is truth and righteousness (that is the father’s teaching) equality is also the nature of the Black woman because that is her limitation. She cannot exceed it. She can only be elevated by god to her fullest quality, by her uniting with god to bring forth the understanding of the culture which the baby is. The making of Islam. 7) GOD – is the Supreme Being, the Blackman from Asia. He is the Universal Builder of all things in existence, who has the power to build or destroy. God is the black man who has knowledge of himself and all things around him. It is the black man accepting and living in the realm of reality about his true nature, which is divine, because God’s cipher is divine. I being God my number is Seven. Seven is perfection. Seven days in a week, seven colors in a rain bow, seven notes in a musical scale, seven seas in the world, seven holes in your head, and there is many more. 8) BUILD or DESTROY – Build is to add on to life step by step to form a positive establishment between the righteous to develop a high elevation of Supreme Understanding by teaching the truth which is Islam. Build is a constant elevation of truth, knowledge and life itself. Destroy is to eliminate or destroy any and all negativity that enters my cipher of supreme harmony. Build is to add and build on to the growth and development of the 5 percent nation of Islam, mentally as well as physically All or any opposing forces must and will be crushed. to build is to elevate the mentalities of self and others. To build, you must start from the root which is the knowledge foundation, and add on to the highest peak. To build is to utilize the supreme powers of the mind which are also able to destroy all negativity (impurities and unrighteousness) to show and prove the power of the universal builder or destroyer. He, who is the wise man of the east. Supreme being, Black man Allah who’s purified state of mind is divided knowledge which makes the culture a divine culture Islam equal I-stimulate-Life-and-matter. Build is to add data to facts making knowledge which is a concrete foundation for positive thoughts. Thus destroying all negativity of the cipher. Now it’s time to build upon Islam positively so we can destroy bad things, which will bring devilishment to Islam. 9) BORN – Is to be brought into existence physically, as well as, mentally. Physically through the mother’s womb, and mentally through Allah’s mathematics. Born is to be brought from being a slave of a mental power and death into the illuminating light of realization about self, through the teachings of a true and living god. Born is knowledge and understanding being manifested through god’s wisdom. (wise words). Born is the completion of all things in existence. All things must come from the knowledge to be born to the cipher. To be born to knowledge of self is to be- born to the reality of one’s self. Knowledge Black man plus wisdom Black woman equal born understanding Black child which is complete within the cipher. Just as it takes 9 months for a black child to be born. It take 9 steps through Allah’s mathematics to be born. 0) CIPHER – is a circle, a completion of 360 degrees, consisting of 120 degrees Knowledge, 120 degrees Wisdom, and 120 degrees Understanding. The cipher has no beginning or ending point so that we may rid our cipher of all negativity. A cipher is a person, place or thing. Everything in life pertains to a cipher. cipher is See-I-Power-Her. PEACE = Positive – Energy – Always – Creates – Elevation. When the number ten is manifested, it would be knowledge add on cipher or knowledge add a cipher or knowledge cipher. Knowledge representing the number 1, and Cipher representing the number zero. ## Lesson 5 What two whole numbers does each square root lie between? Be prepared to explain your reasoning. Can we do any better than “between 3 and 4” for (sqrt<12>) ? Explain a way to figure out if the value is closer to 3.1 or closer to 3.9. ### 5.3: Solutions on a Number Line The numbers (x) , (y) , and (z) are positive, and (x^2 = 3) , (y^2 = 16) , and (z^2 = 30) . Expand Image 1. Plot (x) , (y) , and (z) on the number line. Be prepared to share your reasoning with the class. 2. Plot ( ext- sqrt<2>) on the number line. ### Summary In general, we can approximate the values of square roots by observing the whole numbers around it, and remembering the relationship between square roots and squares. Here are some examples: • (sqrt<65>) is a little more than 8, because (sqrt<65>) is a little more than (sqrt<64>) and (sqrt<64>=8) . • (sqrt<80>) is a little less than 9, because (sqrt<80>) is a little less than (sqrt<81>) and (sqrt<81>=9) . • (sqrt<75>) is between 8 and 9 (it’s 8 point something), because 75 is between 64 and 81. • (sqrt<75>) is approximately 8.67, because (8.67^2=75.1689) . Expand Image Description: <p>A number line with the numbers 8 through 9, in increments of zero point 1, are indicated. The square root of 64 is indicated at 8. The square root of 65 is indicated between 8 and 8 point 1, where the square root of 65 is closer to 8 point 1. The square root of 75 is indicated between 8 point 6 and 8 point 7, the square root of 75 is closer to 8 point 7. The square root of 80 is indicated between 8 point 9 and 9, where the square root of 80 is closer to 8 point 9. The square root of 81 is indicated at 9.</p> If we want to find a square root between two whole numbers, we can work in the other direction. For example, since (22^2 = 484) and (23^2 = 529) , then we know that (sqrt<500>) (to pick one possibility) is between 22 and 23. Many calculators have a square root command, which makes it simple to find an approximate value of a square root. ### Glossary Entries An irrational number is a number that is not a fraction or the opposite of a fraction. Pi ( (pi) ) and (sqrt2) are examples of irrational numbers. A rational number is a fraction or the opposite of a fraction. Some examples of rational numbers are: (frac74,0,frac63,0.2, ext-frac13, ext-5,sqrt9) The square root of a positive number (n) is the positive number whose square is (n) . It is also the the side length of a square whose area is (n) . We write the square root of (n) as (sqrt) . For example, the square root of 16, written as (sqrt<16>) , is 4 because (4^2) is 16. (sqrt<16>) is also the side length of a square that has an area of 16. IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/. Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics. This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information. ## Math Achievement Certificate - VA681, Pack of 30, 8.5" x 11" Math Achievement Certificate features a bright orange background with different math symbols scattered throughout the background. These beautifully designed Hayes certificates are a great way to reward students who have been excelling in math. Teachers and administrators can give these achievement certificates out at award ceremonies and end of the year parties. These student certificates also work great as classroom incentives. They are printed in full color on high quality paper. Certificate reads: Mathematics Achievement Certificate This Honor Is Bestowed Upon _____ in recognition of outstanding achievement in _____ Awarded at _____ this _____ day of _____ year of _____. This math achievement award can be personalized with the student's name and other information pertaining to the award. They can be completed by handwriting the information or printing it. Downloadable template available. â€¢ Inkjet or laser printer compatible â€¢ Certificates measure 8.5" x 11" â€¢ 30 per pack â€¢ Eco-friendly â€¢ High quality matte certificates â€¢ Best Seller ## 8.5: Summary - Mathematics 5 When Jesus returned to Capernaum, a Roman officer [a] came and pleaded with him, 6 “Lord, my young servant [b] lies in bed, paralyzed and in terrible pain.” 7 Jesus said, “I will come and heal him.” 8 But the officer said, “Lord, I am not worthy to have you come into my home. Just say the word from where you are, and my servant will be healed. 9 I know this because I am under the authority of my superior officers, and I have authority over my soldiers. I only need to say, ‘Go,’ and they go, or ‘Come,’ and they come. And if I say to my slaves, ‘Do this,’ they do it.” 10 When Jesus heard this, he was amazed. Turning to those who were following him, he said, “I tell you the truth, I haven’t seen faith like this in all Israel! 11 And I tell you this, that many Gentiles will come from all over the world—from east and west—and sit down with Abraham, Isaac, and Jacob at the feast in the Kingdom of Heaven. 12 But many Israelites—those for whom the Kingdom was prepared—will be thrown into outer darkness, where there will be weeping and gnashing of teeth.” 13 Then Jesus said to the Roman officer, “Go back home. Because you believed, it has happened.” And the young servant was healed that same hour. ## Best Math Courses for Machine Learning ### 1. Mathematics for Machine Learning Specialization Rating- 4.4/5 Provider- Imperial College London Time to Complete- 4 Months (4 hours/week) This is one of the best specialization programs that covers all mathematical topics required for Machine Learning. The aim of this specialization program is to fill the gap and build an intuitive understanding of mathematics. This specialization program is a 3-course series. In the first course, you will learn Linear Algebra, vectors, matrices, and how it relates to data. The Second Course of this specialization is Multivariate Calculus. In this course, you will get a deeper understanding of how to optimize fitting functions to get good fits to data. The last course of this specialization program is Dimensionality Reduction with Principal Component Analysis. This course uses the mathematics from the first two courses to compress high-dimensional data. Along with that, you will work on different real-world projects. After completing this specialization program, you will have gained the prerequisite mathematical knowledge to continue your Machine Learning journey. Let’s see the details of the courses- #### Courses Include- Now, let’s see what skills you will gain after completing this program- #### Skills Gain- • Eigenvalues And Eigenvectors • Principal Component Analysis (PCA) • Multivariate Calculus • Linear Algebra • Basis (Linear Algebra) • Transformation Matrix • Linear Regression • Vector Calculus • Gradient Descent • Dimensionality Reduction • Python Programming #### Extra Benefits- • You will get a Shareable Certificate and Course Certificates upon completion. • Along with that, you will get Course Videos & Readings, Practice Quizzes, Graded Assignments with Peer Feedback, Graded Quizzes with Feedback, Graded Programming Assignments. Now, let’s see whether you should enroll in this specialization program or not? #### You Should Enroll if- • You have High school level maths knowledge. And basic knowledge of Python and NumPy is required for Course 3. #### Interested to Enroll? If yes, then check out all details here- Mathematics for Machine Learning Specialization ### 2. Mathematics for Data Science Specialization Rating- 4.4/5 Provider- National Research University Higher School of Economics Time to Complete- 6 months (4 hours/week) This is another mathematics specialization program, that covers all required math topics for Machine Learning and Data Science. In this specialization, you will learn Discrete Mathematics, Calculus, Linear Algebra, and Probability. This specialization covers a wide range of mathematical tools. In this specialization, you will work on projects like- analyze social graphs, predict estate prices, and uncover hidden relations in the data. There are 4 Courses in this Specialization. Let’s see the details of the courses- #### Courses Include- #### Extra Benefits- • You will get a Shareable Certificate and Course Certificates upon completion. • Along with that, you will get Course Videos & Readings, Practice Quizzes, Graded Assignments with Peer Feedback, Graded Quizzes with Feedback, Graded Programming Assignments. Now, let’s see whether you should enroll in this specialization program or not? #### You Should Enroll if- • You have High schoollevel math, and basic programming in python. #### Interested to Enroll? If yes, then check out all details here- Mathematics for Data Science Specialization ### 3. Data Science Math Skills Rating- 4.5/5 Provider- Duke University Time to Complete- 13 hours This course is offered by Duke University. In this course, you will master the vocabulary, notation, concepts, and algebra rules. In this course, you will learn the following topics- • Set theory, including Venn diagrams • Properties of the real number line • Interval notation and algebra with inequalities • Uses for summation and Sigma notation • Math on the Cartesian (x,y) plane, slope, and distance formulas • Graphing and describing functions and their inverses on the x-y plane • The concept of instantaneous rate of change and tangent lines to a curve • Exponents, logarithms, and the natural log function. • Probability theory, including Bayes’ theorem. #### Extra Benefits- Now, let’s see whether you should enroll in this Course or not? #### You Should Enroll if- • You have no prior knowledge of Math and want to learn basics of Math required for Machine Learning and Data Science. #### Interested to Enroll? If yes, then check out all details here- Data Science Math Skills ### 4. Introduction to Calculus Rating- 4.8/5 Provider– The University of Sydney Time to Complete- 51 Hours This full course is dedicated to Calculus. In this course, you will get a complete understanding of Calculus. This course will teach you key ideas and historical motivation for calculus, while at the same time striking a balance between theory and application. In this course, you will gain familiarity with key ideas of precalculus, develop fluency with the preliminary methodology of tangents and limit, and develop and practice methods of differential calculus and integral calculus. #### Extra Benefits- Now, let’s see whether you should enroll in this Course or not? #### You Should Enroll if- #### Interested to Enroll? If yes, then check out all details here- Introduction to Calculus ### 5. Probabilistic Graphical Models Specialization Rating- 4.6/5 Provider- Stanford University Time to Complete- 4 Months ( 11 hours/week) This Specialization will master you the fundamentals of probabilistic graphical models. Probabilistic graphical models (PGMs) are a rich framework for encoding probability distributions over complex domains. In this specialization program, you will learn the following concepts- • Inference • Bayesian Network • Belief Propagation • Graphical Model • Markov Random Field • Gibbs Sampling • Markov Chain Monte Carlo (MCMC) • Algorithms • Expectation-Maximization (EM) Algorithm This Specialization program is of 3 course series. Let’s see course details- #### Courses Include- #### Extra Benefits- • You will get a Shareable Certificate and Course Certificates upon completion. • Along with that, you will get Course Videos & Readings, Practice Quizzes, Graded Assignments with Peer Feedback, Graded Quizzes with Feedback, Graded Programming Assignments. Now, let’s see whether you should enroll in this specialization program or not? #### You Should Enroll if- • You have some mathematical skills and you are able to program in at least one programming language. • And some previous exposure to basic concepts in discrete probability theory (independence, conditional independence, and Bayes’ rule) will also help. #### Interested to Enroll? If yes, then check out all details here- Probabilistic Graphical Models Specialization ### 6. Statistics with R Specialization Rating- 4.6/5 Provider- Duke University Time to Complete- 7 Months (3 hours/week) This specialization program will give you more in-depth knowledge in Statistics with the help of R. In this program, you will learn how to analyze and visualize data in R and create reproducible data analysis reports, and much more. R is much better than Python for performing statistical operations. So, if you want to master Statistics, then I would recommend this specialization program. This specialization program contains 5 Courses. Let’s see courses details- #### Courses Include- Now, let’s see what statistic skills you will gain after completing this specialization program- #### Skills Gain- • Bayesian Statistics • Linear Regression • Statistical Inference • R Programming • Statistics • Rstudio • Exploratory Data Analysis • Statistical Hypothesis Testing • Regression Analysis • Bayesian Linear Regression • Bayesian Inference • Model Selection #### Extra Benefits- • You will get a Shareable Certificate and Course Certificates upon completion. • Along with that, you will get Course Videos & Readings, Practice Quizzes, Graded Assignments with Peer Feedback, Graded Quizzes with Feedback, Graded Programming Assignments. Now, let’s see whether you should enroll in this specialization program or not? #### You Should Enroll if- • You have basic math knowledge. No previous programming knowledge is required for this course. #### Interested to Enroll? If yes, then check out all details here- Statistics with R Specialization ### 7. Probability and Statistics Rating- 4.6/5 Provider- University of London Time to Complete- 18 Hours This course is especially dedicated to Probability and Statistics. In this course, you will learn many useful tools to deal with uncertainty. The main topics of the course are quantifying uncertainty with probability, descriptive statistics, point and interval estimation of means and proportions, the basics of hypothesis testing, and a selection of multivariate applications. #### Extra Benefits- Now, let’s see whether you should enroll in this Course or not? #### You Should Enroll if- #### Interested to Enroll? If yes, then check out all details here- Probability and Statistics ### 8. Mathematical Foundation For Machine Learning and AI Rating- 4.0/5 Provider- Eduonix Learning Solutions Time to Complete- 4.5 hours This course covers three main mathematical theories: Linear Algebra, Multivariate Calculus, and Probability Theory. This course will refresh your mathematical concepts for AI and Machine Learning. You will learn the following topics in this course- Linear Algebra – • Scalars, Vectors, Matrices, Tensors • Matrix Norms • Special Matrices and Vectors • Eigenvalues and Eigenvectors Multivariate Calculus – • Derivatives • Integrals • Gradients • Differential Operators • Convex Optimization Probability Theory – • Elements of Probability • Random Variables • Distributions • Variance and Expectation • Special Random Variables #### Extra Benefits- • You will get a Certificate of completion and 6 downloadable resources. Now, let’s see whether you should enroll in this Course or not? #### You Should Enroll if- • You want to refresh or learn the mathematical tools required for AI and machine learning. #### Interested to Enroll? If yes, then check out all details here- Mathematical Foundation For Machine Learning and AI So, that’s all. These are 8 Best Math Courses for Machine Learning. Now, it’s time to wrap up. #### Conclusion I hope these courses will give you a strong understanding of math concepts. My aim is to provide you the best resources for Learning. If you have any doubt or questions, feel free to ask me in the comment section. Summary of the Best Math Courses for Machine Learning  S/N Course Name Rating Provider Time to Complete 1. Mathematics for Machine Learning Specialization 4.4/5 Imperial College London 4 Months 2. Mathematics for Data Science Specialization 4.4/5 National Research University Higher School of Economics 6 months 3. Data Science Math Skills 4.5/5 Duke University 13 hours 4. Introduction to Calculus 4.8/5 The University of Sydney 51 Hours 5. Probabilistic Graphical Models Specialization 4.6/5 Stanford University 4 Months 6.. Statistics with R Specialization 4.6/5 Duke University 7 Months 7. Probability and Statistics 4.6/5 University of London 18 Hours 8. Mathematical Foundation For Machine Learning and AI 4.0/5 Eduonix Learning Solutions 4.5 hours Tell me in the comment section, which is the Best Math Course for Machine Learning? #### FAQ What kind of math is needed for machine learning? For Machine Learning, knowledge of Probability, Statistics, Linear Algebra, Calculus, and Matrix are required. Should I learn math before machine learning? In short Yes!. Why? because in order to understand the concepts of Machine Learning Algorithms, you should have math knowledge. That’s why you should learn math first before moving to ML concepts. ## Foundations (161 articles) Edit 1. Foundations of mathematics ### Mathematical logic (32 articles) Edit 1. Mathematical logic (Level 4) 1. Axiom (Level 4) 1. Axiom schema 1. Atomic formula 1. Boolean algebra (Level 4) 1. Karnaugh map 2. Truth table 1. Material conditional 2. Sheffer stroke 1. Free variables and bound variables 2. Quantifier (logic) 1. Existential quantification 2. Universal quantification 1. Skolem normal form 1. Analytical hierarchy 2. Post's theorem ### Set theory (36 articles) Edit 1. Set theory (Level 4) 1. Naive set theory (Level 4) 1. Russell's paradox (Level 4) 1. Complement 2. Element (mathematics) 3. Empty set (Level 4) 4. Intersection (Level 4) 5. Partition of a set 6. Subset (Level 4) 7. Union (Level 4) 8. Universe (mathematics) 1. Cartesian product 2. Ordered pair 1. Multiset 2. Pointed set 1. Reflexive relation 2. Symmetric relation 3. Transitive relation 4. Equivalence relation (Level 4) 1. Indexed family 1. Aleph number 2. Cantor's theorem (Level 4) 1. Cantor's diagonal argument 1. Forcing (mathematics) 1. Transfinite induction 2. Von Neumann universe ### Axiomatic systems (19 articles) Edit 1. Axiomatic system 2. Zermelo–Fraenkel set theory 1. Axiom of extensionality 2. Axiom of regularity 3. Axiom schema of specification 4. Axiom of pairing 5. Axiom of union 6. Axiom schema of replacement 7. Axiom of infinity 8. Axiom of power set 1. Banach–Tarski paradox 2. Well-ordering theorem 3. Zorn's lemma 1. Mathematical induction (Level 4) ### Computability theory (26 articles) Edit 1. Computability theory (Level 4) 1. Church–Turing thesis 1. Combinatory logic 2. Lambda calculus 3. Turing machine 1. General recursive function 1. Primitive recursive function 2. μ operator 1. Gödel's incompleteness theorems (Level 4) 2. UTM theorem 1. Entscheidungsproblem 1. Halting problem 2. Rice's theorem 3. Turing's proof 4. Universal Turing machine 1. Computable set 2. Computably inseparable 1. Turing reduction 1. Many-one reduction 2. Turing degree ### Model theory (19 articles) Edit 1. Model theory 1. Compactness theorem 2. Gödel's completeness theorem 3. Löwenheim–Skolem theorem 1. Skolem's paradox 1. Elementary equivalence 2. Interpretation (model theory) 3. Reduct 4. Saturated model 5. Signature (logic) 6. Transfer principle 7. Type (model theory) 8. Ultraproduct 1. Amalgamation property 2. Model complete theory 3. Quantifier elimination 4. Stable theory ### Category theory (28 articles) Edit 1. Category theory (Level 4) 1. Commutative diagram 1. Five lemma 2. Nine lemma 3. Snake lemma 1. Cartesian closed category 2. Comma category 3. Concrete category 4. Opposite category 5. Subcategory 1. Adjoint functors 2. Forgetful functor 3. Hom functor 4. Yoneda lemma 1. Free object 2. Limit (category theory) 1. Coproduct 2. Initial and terminal objects 3. Kan extension 4. Product (category theory) 5. Pullback (category theory) 6. Pushout (category theory) ## Self Check Problem The following dollar amounts were the hourly collections from a Salvation Army kettle at a local store one day in December:$19, $26,$25, $37,$32, $28,$22, $23,$29, $34,$39, and \$31. Determine the range and interquartile range for the amount collected.

Definition: The five-number summary of a data set consists of the five numbers determined by computing the minimum, Q1 , median, Q3 , and maximum of the data set.

Example 1: Find the five-number summary for the data set <3, 7, 8, 5, 12, 14, 21, 13, 18>.

From our Example 1's on the previous pages, we see that the five-number summary is:
Minimum: 3 Q1 : 6 Median: 12 Q3 : 16 Maximum: 21

Example 2: Find the five-number summary for the data set <3, 7, 8, 5, 12, 14, 21, 15, 18, 14>.

From our Example 2's on the previous pages, we see that the five-number summary is:

Minimum: 3 Q1 : 7 Median: 13 Q3 : 15 Maximum: 21

One way to describe what is typical or characteristic for a data set is by looking at the center and spread of its distribution.

Let’s compare the distribution of cat weights and dog weights shown on these dot plots.

Expand Image

Description: <p>A dot plot for "cat weights in kilograms". The numbers 2 through 12 are indicated. The data are as follows: 3 kilograms, 2 dots. 3.5 kilograms, 3 dots. 4 kilograms, 4 dots. 4.5 kilograms, 5 dots. 5 kilograms, 5 dots. 5.5 kilograms, 4 dots. 6 kilograms, 3 dots. 6.5 kilograms, 3 dots. 7 kilograms, 1 dot.</p>

Expand Image

Description: <p>A dot plot for "dog weights in kilograms". The numbers 2 through 12 are indicated. The data are as follows: 5 kilograms, 1 dot. 5.5 kilograms, 2 dots. 6 kilograms, 2 dots. 6.5 kilograms, 3 dots. 7 kilograms, 4 dots. 7.5 kilograms, 4 dots. 8 kilograms, 3 dots. 8.5 kilograms, 3 dots. 9 kilograms, 3 dots. 9.5 kilograms, 2 dots. 10 kilograms, 2 dots. 10.5 kilograms, 1 dot. 11 kilograms, 1 dot.</p>

The collection of points for the cat data is further to the left on the number line than the dog data. Based on the dot plots, we may describe the center of the distribution for cat weights to be between 4 and 5 kilograms and the center for dog weights to be between 7 and 8 kilograms.

We often say that values at or near the center of a distribution are typical for that group. This means that a weight of 4–5 kilograms is typical for a cat in the data set, and weight of 7–8 kilograms is typical for a dog.

We also see that the dog weights are more spread out than the cat weights. The difference between the heaviest and lightest cats is only 4 kilograms, but the difference between the heaviest and lightest dogs is 6 kilograms.

A distribution with greater spread tells us that the data have greater variability. In this case, we could say that the cats are more similar in their weights than the dogs.

In future lessons, we will discuss how to measure the center and spread of a distribution.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.