The ratio of width to height of standard-definition television.

In mathematics, a ratio is a relationship between two numbers of the same kind^[1] (e.g., objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two^[2] which explicitly indicates how many times the first number contains the second (not necessarily an integer).^[3] In layman's terms a ratio represents, simply, for every amount of one thing, how much there is of another thing. For example, supposing one has 10 pairs of socks for every pair of shoes, the ratio of shoes:socks would be 1:10 and the ratio of socks:shoes would be 10:1.

Notation and terminology[link]

The ratio of numbers A and B can be expressed as:^[4]

• the ratio of A to B
• A is to B
• A:B
• A rational number which is the quotient of A divided by B

The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.

The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. this latter form, when spoken or written in the English language, is often expressed as

A is to B as C is to D.

Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.^[5]

Ratios are sometimes used with three or more terms. The dimensions of a two by four that is ten inches long are 2:4:10.

History and etymology[link]

Look up ratio in Wiktionary, the free dictionary.

It is impossible to trace the origin of the concept of ratio, since the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society.^[6] However, it is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). (A rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.^[7] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.^[8]

Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.^[9] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.^[10]

The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.^[11]

Euclid's definitions[link]

Book V of Euclid's Elements has 18 definitions, all of which relate to ratios.^[12] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a part of a quantity is another quantity which "measures" it and conversely, a multiple of a quantity is another quantity which it measures. In modern terminology this means that a multiple of a quantity is that quantity multiplied by an integer greater than one and a part of a quantity (meaning aliquot part) is that which, when multiplied by an integer greater than one, gives the quantity. Euclid does not define the term "measure" as used here but one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second. Note that these definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.

Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself.^[13] Euclid defines a ratio to be between two quantities of the same type, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each which exceeds the other. In modern notation, a ratio exists between quantities p and q if there exist integers m and n so that mp>q and nq>m. This condition is known as the Archimedean property.

Definition 5 is the most complex and difficult; it defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of the quotients of incommensurables, so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another. Though it may not be possible to assign a rational value to a ratio, it is possible to compare a ratio with a rational number. Specifically, given two quantities, p and q, and a rational number m/n we can say that the ratio of p to q is less than, equal to, or greater than m/n when np is less than, equal to, or greater than mq respectively. Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number. In modern notation this says that given quantities p, q, r and s, then p:q::r:s if for any positive integers m and n, np<mq, np=mq, np>mq according as nr<ms, nr=ms, nr>ms respectively. There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers.^[14]

Definition 6 says that quantities that have the same ratio are proportional or in proportion. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".

Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities p, q, r and s, then p:q>r:s if there are positive integers m and n so that np>mq and nr≤ms.

As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p, q and r to be in proportion when p:q::q:r. This is extended to 4 terms p, q, r and s as p:q::q:r::r:s, and so on. Sequences which have the property that the ratios of consecutive terms are equal are called Geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and if p, q, r and s are in proportion then p:s is the triplicate ratio of p:q. If p, q and r are in proportion then q is called a mean proportional to (or the geometric mean of) p and r. Similarly, if p, q, r and s are in proportion then q and r are called two mean proportionals to p and s.

Examples[link]

The quantities being compared in a ratio might be physical quantities such as speed or length, or numbers of objects, or amounts of particular substances. A common example of the last case is the weight ratio of water to cement used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made. Equivalently it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement..

Older televisions have a 4:3 "aspect ratio", which means that the width is 4/3 of the height; modern widescreen TVs have a 16:9 aspect ratio.

Fraction[link]

If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason.

Number of terms[link]

In general, when comparing the quantities of a two-quantity ratio, this can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount/size/volume/number of the first quantity will be Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{2}{3}

that of the second quantity. This pattern also works with ratios with more than two terms. However, a ratio with more than two terms cannot be completely converted into a single fraction; a single fraction represents only one part of the ratio since a fraction can only compare two numbers.  If the ratio deals with objects or amounts of objects, this is often expressed as "for every two parts of the first quantity there are three parts of the second quantity".

Percentage ratio[link]

If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the lowest common denominator, or to express them in parts per hundred (percent).

If a mixture contains substances A, B, C & D in the ratio 5:9:4:2 then there are 5 parts of A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, this is converted to percentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10).

Proportions[link]

If the two or more ratio quantities encompass all of the quantities in a particular situation, for example two apples and three oranges in a fruit basket containing no other types of fruit, it could be said that "the whole" contains five parts, made up of two parts apples and three parts oranges. In this case, Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{2}{5} , or 40% of the whole are apples and Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{3}{5} , or 60% of the whole are oranges. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.

Reduction[link]

Note that ratios can be reduced (as fractions are) by dividing each quantity by the common factors of all the quantities. This is often called "cancelling." As for fractions, the simplest form is considered to be that in which the numbers in the ratio are the smallest possible integers.

Thus, the ratio 40:60 may be considered equivalent in meaning to the ratio 2:3 within contexts concerned only with relative quantities.

Mathematically, we write: "40:60" = "2:3" (dividing both quantities by 20).

Grammatically, we would say, "40 to 60 equals 2 to 3."

An alternative representation is: "40:60::2:3"

Grammatically, we would say, "40 is to 60 as 2 is to 3."

A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms.

Sometimes it is useful to write a ratio in the form 1:n or n:1 to enable comparisons of different ratios.

For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4)

Alternatively, 4:5 can be written as 0.8:1 (dividing both sides by 5)

Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the colon, though, mathematically, this makes it a factor or multiplier.

Dilution ratio[link]

Ratios are often used for simple dilutions applied in chemistry and biology. A simple dilution is one in which a unit volume of a liquid material of interest is combined with an appropriate volume of a solvent liquid to achieve the desired concentration. The dilution factor is the total number of unit volumes in which your material will be dissolved. The diluted material must then be thoroughly mixed to achieve the true dilution. For example, a 1:5 dilution (verbalize as "1 to 5" dilution) entails combining 1 unit volume of solute (the material to be diluted) + 4 unit volumes (approximately) of the solvent to give 5 units of the total volume. (Some solutions and mixtures take up slightly less volume than their components.)

The dilution factor is frequently expressed using exponents: 1:5 would be 2e−1 (5⁻¹ i.e. one-fifth:one); 1:100 would be 10e−2 (10⁻² i.e. one hundredth:one), and so on.

There is often confusion between dilution ratio (1:n meaning 1 part solute to n parts solvent) and dilution factor (1:n+1) where the second number (n+1) represents the total volume of solute + solvent. In scientific and serial dilutions, the given ratio (or factor) often means the ratio to the final volume, not to just the solvent. The factors then can easily be multiplied to give an overall dilution factor.

In other areas of science such as pharmacy, and in non-scientific usage, a dilution is normally given as a plain ratio of solvent to solute.

Odds[link]

Odds (as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. On the other hand, the probability of success is 30%. In every ten trials, there are three wins and seven losses.

Different units[link]

Ratios are unitless when they relate quantities which have units of the same dimension.

For example, the ratio 1 minute : 40 seconds can be reduced by changing the first value to 60 seconds. Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2.

In chemistry, mass concentration "ratios" are usually expressed as w/v percentages, and are really proportions.

For example, a concentration of 3% w/v usually means 3g of substance in every 100mL of solution. This cannot easily be converted to a pure ratio because of density considerations, and the second figure is the total amount, not the volume of solvent.

See also[link]

• Proportionality (mathematics)
• Aspect ratio
• Fraction (mathematics)
• Interval (music)
• Rule of three (mathematics)
• Price/performance ratio
• Golden ratio

References[link]

1. ^ Wentworth, p. 55
2. ^ New International Encyclopedia
3. ^ Penny Cyclopedia, p. 307
4. ^ New International Encyclopedia
5. ^ New International Encyclopedia
6. ^ Smith, p. 477
7. ^ Penny Cyclopedia, p. 307
8. ^ Smith, p. 478
9. ^ Heath, p. 112
10. ^ Heath, p. 113
11. ^ Smith, p. 480
12. ^ Heath, reference for section
13. ^ "Geometry, Euclidean" Encyclopædia Britannica Eleventh Edition p682.
14. ^ Heath p. 125

Further reading[link]

• "Ratio" The Penny Cyclopædia vol. 19, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
• "Proportion" New International Encyclopedia, Vol. 19 2nd ed. (1916) Dodd Mead & Co. pp270-271
• "Ratio and Proportion" Fundamentals of practical mathematics, George Wentworth, David Eugene Smith, Herbert Druery Harper (1922) Ginn and Co. pp. 55ff
• The thirteen books of Euclid's Elements, vol 2. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. pp. 112ff. http://books.google.com/books?id=lxkPAAAAIAAJ&pg=RA2-PA112. 
• D.E. Smith, History of Mathematics, vol 2 Dover (1958) pp. 477ff

Barbara Lee Fredrickson (born June 15, 1964)^[1] is an American professor in the department of psychology at the University of North Carolina at Chapel Hill, where she is the Kenan Distinguished Professor of Psychology.

Fredrickson is a social psychologist who conducts research in emotions and positive psychology. Her main work is related to her broaden-and-build theory of positive emotions, which suggests that positive emotions lead to novel, expansive, or exploratory behavior, and that, over time, these actions lead to meaningful, long-term resources such as knowledge and social relationships. She is the author of Positivity, a general-audience book that draws on her own research and that of other social scientists.

Fredrickson earned her Ph.D. from Stanford University in 1990. She was a professor at the University of Michigan for 10 years before moving to the University of North Carolina at Chapel Hill.

Fredrickson's mentors include Robert Levenson and Laura Carstensen.

Research[link]

Broaden-and-build[link]

Studies from Fredrickson's lab have randomly-assigned participants watch films that induce positive emotions such as amusement and contentment, negative emotions such as fear and sadness, or no emotions. Compared to people in the other conditions, participants who experience positive emotions show heightened levels of creativity, inventiveness, and "big picture" perceptual focus. Longitudinal studies show that positive emotions play a role in the development of long-term resource such as psychological resilience and flourishing.^[2]

The undoing effect[link]

Fredrickson and others hypothesize that positive emotions undo the cardiovascular effects of negative emotions. When people experience stress, they show increased heart rate, higher blood sugar, immunosuppression, and other adaptations optimized for immediate action. If individuals do not regulate these changes once the stress is past, they can lead to illness, such as coronary disease, and heightened mortality. Both lab research and survey research indicate that positive emotions help people who were previously under stress relax back to their physiological baseline.^[3]

Sex differences in self-objectification[link]

Prior to her work on positive emotions, Fredrickson researched social and environmental cues that can carry sexist messages and enhance stereotypical gender differences. She found that when women are randomly assigned dress in a way that calls attention to their bodies, they show impaired performance on a math task and were literally more likely to "throw like a girl". This research suggested that drawing attention to women's bodies also activated stereotypical beliefs about their gender.^[4]

Recognition[link]

Fredrickson received the Templeton Prize in Positive Psychology in 2000 for her work on the broaden-and-build theory, which included a $100,000 grant to fund her work. Her work has been supported by grants from the National Institute of Mental Health.

Selected publications[link]

• Fredrickson, Barbara (2009). Positivity. New York: Crown.

References[link]

External links[link]

• Fredrickson Lab website
• Publications by Fredrickson and colleagues

Dr. John Gottman with his wife Julie

John Mordecai Gottman (born April 26, 1942) is a Professor emeritus in psychology known for his work on marital stability and relationship analysis through scientific direct observations published in peer-reviewed literature. The lessons derived from this work represent a partial basis for the relationship counseling movement which aim to improve relationship functioning and the avoidance of those behaviors shown by Gottman and other researchers to harm human relationships.^[1] Dr. Gottman is a Professor Emeritus of psychology at the University of Washington. With his wife Dr. Julie Schwartz, Gottman heads a non-profit research institute (The Relationship Research Institute) and a for-profit therapist training entity (The Gottman Institute).^[2]

Predictions of divorce[link]

Gottman developed multiple models scales and formulas to predict marital stability and divorce in couples, and has completed seven studies in this field.^[3]  The predictive studies regarding newlywed couples are most well known.

1992 Study[link]

The original study was conducted by Gottman and Buehlman’s in 1992, in which they interviewed couples with children, and an a posteriori modeling yielded a discriminant function which discriminate who divorces with 94% accuracy.^[4]  However, he believed that since early married life is a period of change and adjustment, and perceptions are being formed, Gottman sought to predict marital stability and divorce through couples’ perceptions during the first year of marriage.^[5]

1998 Study[link]

In a 1998 study, Gottman developed a model to predict which newlywed couples will remain married and which will divorce four to six years later. He claims that his model has 90% accuracy. He also claims 81% percent accuracy for another model about which marriages survived after seven to nine years.^[6]

2000 Study[link]

Gottman’s follow-up study published in 2000 with newlywed couples used the Oral History Interview to predict marital stability and divorce. Gottman had 87.4% accuracy in predicting marriage or divorce within the couples’ first 5 years of marriage. He used couples’ perceptions about their marriage and each other to predict marital stability or divorce.^[7]

Method[link]

Gottman’s predictions are based on perceived marital bond. In his 2000 study, Gottman conducted oral interviews with 95 newlywed couples. Couples were asked about their relationship, mutual history, and philosophy towards marriage. The interview measured the couple's perceptions of their history and marriage by focusing on the positive or negative qualities of the relationship expressed in the telling of the story. Rather than scoring the content of their answers, interviewers used the Oral History Interview coding system, developed by Buehlman and Gottman in 1996, to measure spouses' perceptions about the marriage and about each other. Therefore, the couples’ perception was used to predict marital stability or divorce. The more positive their perceptions and attitudes were about their marriage and each other, the more stable the marriage.^[8]  

His models partly rely on Paul Ekman's method of analyzing human emotion and microexpressions.

Replicate Study[link]

A replicate study published in by Rand D. Conger in 1996 based on couples’ perceptions predicted marital stability and divorce with a 80% accuracy.^[9] 

Critique[link]

A paper by Richard E. Heyman, "The hazard of predicting divorce without Cross validation"[1] analyzes 15 divorce prediction models and questions their validity for three reasons. 1) when analyzing a given dataset, it is possible to overfit the model to the data. Which will work extremely nice for this dataset, but will t work when tested on fresh data.  2) 90% prediction may actually mean much less when considering false positives and the low base rates of divorce. 

The author shows his points by creating a divorce prediction model with a data set, and showing its low validity when the above considerations are tested.

Journalist Laurie Abraham also disputed the prediction power of Gottman's method. Abraham writes, "What Gottman did wasn't really a prediction of the future but a formula built after the couples' outcomes were already known. This isn't to say that developing such formulas isn't a valuable—indeed, a critical—first step in being able to make a prediction. The next step, however—one absolutely required by the scientific method—is to apply your equation to a fresh sample to see whether it actually works. That is especially necessary with small data slices (such as 57 couples), because patterns that appear important are more likely to be mere flukes. But Gottman never did that."^[10] The Gottman Relationship Institute claims that 6 of 7 of Gottman's studies have been properly predictive, by the standard that the predictors, but not their specific relationship to the outcome, were selected in advance.^[11] However, Gottman's 2002 paper makes no claims to accuracy in terms of binary classification, and is instead a regression analysis of a two factor model where skin conductance levels and oral history narratives encodings are the only two statistically significant variables. Facial expressions using Ekman's encoding scheme were not statistically significant.^[12]

Results[link]

Gottman found that the four negative behaviors that most predict divorce are criticism of partners’ personality, contempt (from a position of superiority), defensiveness, and stonewalling, or emotional withdrawal from interaction. On the other hand, stable couples handle conflicts in gentle, positive ways, and are supportive of each other.^[13]  

He developed the Gottman Method Couple’s Therapy based on his research findings. The therapy aimed to increase respect, affection, and closeness, break through and resolve conflict, generate greater understandings, and to keep conflict discussions calm.^[14]  The Gottman Method sought to help couples build a happy and stable marriage.

Contempt and marriage[link]

Gottman's theory states that there are four major emotional reactions that are destructive and thus are the 4 predictors to a divorce: criticism, defensiveness, stonewalling, and contempt. Among these four, Gottman considers contempt the most important of them all.^[15]

Seven Principles[link]

In his book, The Seven Principles of Making Marriage Work, Gottman discusses behaviors that he has observed in marriages that are successful and those that are detrimental to marriage based on his research conducted at his lab in Seattle, Washington. He has outlined seven principles that will reinforce the positive aspects of a relationship and help marriages endure during the rough moments.

1. Enhance Your Love Maps. Gottman defines a love map as the place in your brain where you store information pertaining to your partner. This is crucial in really knowing your partner, their dreams, hopes, interests, and maintaining their interest throughout the relationship.
2. Nurture Your Fondness and Admiration. This means laying down a positive view about your spouse, respecting and appreciating their differences.
3. Turn Toward Each Other Instead of Away. Acknowledging your partner's small moments in life and orienting yourself towards them will maintain that necessary connection that is vital for the relationship.
4. Let Your Partner Influence You. It is important to maintain your own identity in a relationship, but it is equally important to yield to your partner and give in. If both partners allow one another this influence, then they will learn to respect one another on a deeper level.
5. Solve Your Solvable Problems. It is important to compromise on issues that can be resolved, which Gottman believes can be accomplished by these five steps: soften your startup, learn to make and receive repair attempts, soothe yourself and each other, compromise, and be tolerant of each other’s faults.
6. Overcome Gridlock. Major issues that cannot be resolved because both partners’ views are so fundamentally different involves understanding of the other person and deep communication. The goal is to at least get to a position that allows the other person to empathize with the partner's view, even if a compromise cannot be reached.
7. Create Shared Meaning. Create a shared value system that continually connects the partners through rituals/traditions, shared roles and symbols.^[16]

Personal life[link]

John Gottman was born in the Dominican Republic to Orthodox Jewish parents. His father was a rabbi in pre-WWII Vienna. John was educated in a Lubavitch yeshiva elementary school in Brooklyn, and currently identifies with Conservative Judaism.^[17]

Two decades ago, he married Julie Gottman, a psychotherapist. The couple currently lives in Washington state. They have a biological child together, Moriah Gottman.

Awards and honors[link]

Gottman has been the recipient of four National Institute of Mental Health Research Scientist Awards, the American Association for Marriage and Family Therapy Distinguished Research Scientist Award, the American Family Therapy Academy Award for Most Distinguished Contributor to Family Systems Research, the American Psychological Association Division of Family Psychology, Presidential Citation for Outstanding Lifetime Research Contribution and the National Council of Family Relations, 1994 Burgess Award for Outstanding Career in Theory and Research.^[18]

Media appearances[link]

Gottman has been seen on, among other television programs, Good Morning America, the Today Show, the CBS Morning News and Oprah. He has been profiled in the New York Times, the Ladies Home Journal, Redbook, Glamour, Woman’s Day, People, Self, the Reader’s Digest, Psychology Today, the Seattle Times and the Seattle Post-Intelligencer.^[18]

Works[link]

Gottman has published over 190 papers, and is the author or co-author of 40 books, notably:^[18]

• Why Marriages Succeed or Fail…and How You Can Make Yours Last (Simon & Schuster, 1994)
• Raising An Emotionally Intelligent Child: The Heart of Parenting (Simon and Schuster, 1997) – written with Joan Declaire
• The Marriage Clinic (W.W. Norton, 1999), W W Norton page
• The Seven Principles for Making Marriage Work (Crown Publishers, 1999) – a New York Times bestseller
• The Relationship Cure, A 5-Step Guide for Building Better Connections with Family, Friends, and Lovers (Crown Publishers, 2001)
• Ten Lessons to Transform Your Marriage: America's Love Lab Experts Share Their Strategies for Strengthening Your Relationship (Crown Publishers, 2006)
• And Baby Makes Three: The Six-Step Plan for Preserving Marital Intimacy and Rekindling Romance After Baby Arrives (Crown Publishers, 2007)
• "The science of trust" about a new concept of truth in relationships that is a crucial factor

References[link]

External links[link]

• The Gottman Institute
• The Mathematics of Love - An interview (Edge)
• An Interview with John Gottman (Psychotherapy.net)
• John Gottman: Raising an Emotionally Intelligent Child KUOW-FM Speaker Forum