This is an idea I have considered for many years. It fits with my overall view of a lot of evidence from treatment studies.
For many actions in life, an event either happens, or it doesn't. This seems obvious, I guess. You either throw a ball, or you don't. You either show up for work, or you don't. (Mind you, in my own case, I would say that my own modest level of athletic skill causes me quite often to "sort of" throw a ball, or "sort of" swim.)
In medicine, many actions are similarly unambiguous. The surgical removal of an appendix either happens, or it doesn't. An infection either responds to an antibiotic, or it doesn't. Clear.
Yet, I find that many treatments in medicine are much less clear.
In the case of psychiatric treatments, it has been a theory of mine that the drug (or therapy) may reduce the probability of a symptom occurring, in addition to, or instead of, directly reducing the symptom (or not). This phenomenon may be apparent not only in studies of populations, but in an individual.
Many disease processes are driven by multiple variables, which, together, alter probabilities of symptom recurrence. The variables may include genetic factors, environmental stress, etc. There may be a core phenomenon in nature, as manifest on a chemical, or even quantum-mechanical, level, of minute, truly random events, influencing a cascade of effects. The presence of a medication in the body may be just one more variable, influencing the likelihood of a symptom occuring, or developing, or advancing.
Some medications may appear not to be working, if a short-term view is taken. But in a longer-term view, it may be seen that symptom frequency and intensity is diminished. This is consistent with the theory that the medication affects probabilities.
This theory supports the idea that medications, and other psychiatric treatments, could have an important preventative role, beyond their role in an acute situation. And it encourages giving treatments a long period of time to work--at least months, if not years-- in order to most accurately assess effectiveness.
There have been some long-term studies which support this idea, but unfortunately most of the treatment studies in psychiatry have been relatively short-term (only a few months of follow-up, rarely more than a year).
Tuesday, February 10, 2009
Omega-3 Supplementation
Omega-3 fatty acids are present in a variety of foods.
The fatty acids EPA and DHA are present mainly in fish such as salmon, herring, mackerel, anchovies, and sardines. These fatty acids, especially DHA, are probably important for brain function, and are also found in the retina of the eye.
Another omega-3 fatty acid, ALA, is present from plant sources such as canola oil, flax, and walnuts. ALA may be converted in the body to DHA.
There is some evidence that there are health benefits from diets higher in omega-3 fatty acids, or diets supplemented with extra omega-3.
Of interest for psychiatry, omega-3 supplementation may be a safe adjunct in the treatment of depression. Fish oil is probably the simplest source of extra EPA and DHA.
The only problem with increasing fish consumption is the exposure to environmental contaminants such as mercury and PCBs. Fish oil capsules may actually have less of these contaminants than pure fish, especially if the oil has been refined to remove contaminants. In any case, I think the benefit-risk ratio is very favourable, and that 1-3 capsules per day of fish oil is quite safe. And I feel confident to recommend increased fish intake in the diet. For vegetarians, increased intake of walnuts, canola, and flax could be recommended.
http://www.ncbi.nlm.nih.gov/pubmed/18183532
(a review of the studies over the past decade looking at omega-3 supplements in mood disorders)
http://www.ncbi.nlm.nih.gov/pubmed/16741195
(a nice review from The American Journal of Psychiatry in 2006, summarizing epidemiological data associating low fish consumption with higher rates of mood disorder, and summarizing some of the treatment studies showing antidepressant effects of omega-3 supplements in depression, bipolar disorder, and borderline personality)
http://www.ncbi.nlm.nih.gov/pubmed/19156158
(this is a recent study showing beneficial effects of omega-3 supplements in children with bipolar symptoms;but it was not a randomized or controlled study)
http://www.ncbi.nlm.nih.gov/pubmed/19200125
(this is a recent local study analyzing fish oil supplements for environmental pollutant levels, such as PCBs. Based on this study, one should avoid supplements of products such as seal or shark oils, which have much higher contaminant levels.)
http://www.ncbi.nlm.nih.gov/pubmed/19139352
(one of the articles summarizing evidence that omega-3 intake reduces the incidence or progression of macular degeneration, which is a common cause of visual loss in those over 65 years of age).
http://www.ncbi.nlm.nih.gov/pubmed/19064523
(a huge study, published in 2006, involving data from over
40 000 people over 18 years of follow-up--it shows a slight reduction in cardiac disease associated with higher fish consumption, but no change in overall "major chronic disease risk". But, incredibly, and unfortunately, they did not include mood or other psychiatric disorders in their assessment of "chronic disease" outcomes. Yet, studies of this type exemplify that The American Journal of Clinical Nutrition is an excellent journal, a valuable and practical source of evidence-based health information which could guide nutritional choices).
The fatty acids EPA and DHA are present mainly in fish such as salmon, herring, mackerel, anchovies, and sardines. These fatty acids, especially DHA, are probably important for brain function, and are also found in the retina of the eye.
Another omega-3 fatty acid, ALA, is present from plant sources such as canola oil, flax, and walnuts. ALA may be converted in the body to DHA.
There is some evidence that there are health benefits from diets higher in omega-3 fatty acids, or diets supplemented with extra omega-3.
Of interest for psychiatry, omega-3 supplementation may be a safe adjunct in the treatment of depression. Fish oil is probably the simplest source of extra EPA and DHA.
The only problem with increasing fish consumption is the exposure to environmental contaminants such as mercury and PCBs. Fish oil capsules may actually have less of these contaminants than pure fish, especially if the oil has been refined to remove contaminants. In any case, I think the benefit-risk ratio is very favourable, and that 1-3 capsules per day of fish oil is quite safe. And I feel confident to recommend increased fish intake in the diet. For vegetarians, increased intake of walnuts, canola, and flax could be recommended.
http://www.ncbi.nlm.nih.gov/pubmed/18183532
(a review of the studies over the past decade looking at omega-3 supplements in mood disorders)
http://www.ncbi.nlm.nih.gov/pubmed/16741195
(a nice review from The American Journal of Psychiatry in 2006, summarizing epidemiological data associating low fish consumption with higher rates of mood disorder, and summarizing some of the treatment studies showing antidepressant effects of omega-3 supplements in depression, bipolar disorder, and borderline personality)
http://www.ncbi.nlm.nih.gov/pubmed/19156158
(this is a recent study showing beneficial effects of omega-3 supplements in children with bipolar symptoms;but it was not a randomized or controlled study)
http://www.ncbi.nlm.nih.gov/pubmed/19200125
(this is a recent local study analyzing fish oil supplements for environmental pollutant levels, such as PCBs. Based on this study, one should avoid supplements of products such as seal or shark oils, which have much higher contaminant levels.)
http://www.ncbi.nlm.nih.gov/pubmed/19139352
(one of the articles summarizing evidence that omega-3 intake reduces the incidence or progression of macular degeneration, which is a common cause of visual loss in those over 65 years of age).
http://www.ncbi.nlm.nih.gov/pubmed/19064523
(a huge study, published in 2006, involving data from over
40 000 people over 18 years of follow-up--it shows a slight reduction in cardiac disease associated with higher fish consumption, but no change in overall "major chronic disease risk". But, incredibly, and unfortunately, they did not include mood or other psychiatric disorders in their assessment of "chronic disease" outcomes. Yet, studies of this type exemplify that The American Journal of Clinical Nutrition is an excellent journal, a valuable and practical source of evidence-based health information which could guide nutritional choices).
Monday, February 9, 2009
Lithium
I'd like to develop this post gradually, as there is a lot of evidence to summarize and refer to.
But here is a start:
"Mood stabilizers" are drugs which are thought to help treat the symptoms of bipolar disorder. It is hoped that these drugs might reduce manic symptoms, prevent recurrence of manic symptoms, while also reducing or preventing symptoms of depression.
The first treatments for manic episodes were sedatives, including barbiturates and antipsychotics.
The first "mood stabilizer" though, was lithium carbonate.
Lithium itself is the third-simplest element in the universe, after hydrogen and helium. It tends to form salts. It is structurally very similar to sodium, which is a salt-forming element essential to most every life function (that is why we find sodium ions abundantly in all body fluids, and in a similar concentration in the ocean; table salt consists of sodium and chlorine atoms which join together as crystals). Yet, lithium is not normally present in the human body, and is much less common in the universe as well, compared to hydrogen, oxygen, carbon, sodium, etc.
The mechanism of lithium's action in the body, when used as a drug, is still poorly understood. Its similarity to sodium is probably essential to its mechanism. I'll add to this commentary later, but for now I will say that the mechanism of lithium salts probably involves multiple actions inside of nerve cells; these actions may modulate cellular activity.
Here are some of the clinical actions of lithium carbonate, when used as a medication in those with bipolar disorder (I will list the actions in order of how clearly proven and substantial the effect is):
1) It reduces symptoms of mania
2) It may reduce the length of a manic episode (note that just because a treatment reduces symptom severity, it may not reduce the duration of the symptoms).
3) It may prevent the recurrence of manic symptoms
4) It may prevent the recurrence of depressive symptoms
5) It may reduce depressive symptoms when they occur
There are other uses for lithium carbonate as well:
1) It can be combined with an antidepressant to improve symptom control in unipolar depression
2) It may help treat specific symptoms such as irritability and rage
3) It helps prevent cluster headaches (a type of severe, recurrent headache)
Lithium is probably most useful in "classic bipolar disorder", in which individuals experience manic episodes with elevated mood (as opposed to irritable or "dysphoric" mood), and in which the mood episodes are not recurring frequently during an average year (i.e. there is no "rapid cycling").
Here are some of the side-effects of lithium:
1) thirst, increased urine production
2) tremor (shaky hands)
3) nausea
4) sedation -- usually it is a much less pronounced type of sedation compared to antipsychotics, benzodiazepines, or other "sleeping pills". But there can be feelings of reduced energy, reduced clarity of thinking, or lethargy
5) toxicity to the kidneys -- this is not common, but needs to be checked for regularly
6) inhibition of thyroid function -- this is not permanent, nor is it harmful to the thyroid gland (in fact, it may "rest" the thyroid gland); but diminished thyroid levels, if present, requires treatment with a thyroid supplement)
7) acne or other skin rashes
8) toxicity in overdose
There have been a few studies questioning the effectiveness of lithium, particularly in terms of its value in preventing recurrent mood episodes. But for many people it does appear to be very effective, both as an acute treatment and as a preventative agent. It probably works much better as an "anti-manic" agent than an "anti-depressant".
There are various forms of lithium, and various dosing regimes. In most cases, it can be dosed simply: once at bedtime. The concentration of lithium in the blood needs to be measured periodically. Levels which are too high can increase the likelihood of toxicity (mind you, excessive levels could usually be assessed on the basis of side-effect complaints); levels which are too low may not be effective.
In my experience, some people may benefit from staying on lithium, but adjusting the dose to a point that is more tolerable for them. It may not necessarily be true that everyone needs to have a full therapeutic concentration of lithium in order for it to work. For some people, the side effects may outweigh the benefit at full doses.
However, it is a frequent situation in an emergency room, or on a mood disorders hospital ward, that people with clear histories of bipolar disorder, stable on medication, end up having a recurrence of severe mania a few weeks or months after tapering or stopping their medication (often lithium). In some of these cases, the manic symptoms may have already been building up, leading the person to discontinue their medication (rather than the other way around). But in many of these cases, it seems to me that the lithium had been protecting them, and that the recurrence of mania happened because of medication discontinuation.
There is also some evidence that sudden lithium discontinuation can provoke increased mood instability. So, while there are no overt withdrawal symptoms from stopping lithium, it should be tapered slowly if possible (I would say over 1-2 months at least).
It should be emphasized that lithium is not a perfect drug, either in terms of side effects or in terms of effectiveness. Many people on full doses of lithium still experience relapses of mania. But it is quite clear, from decades of experience, that lithium can be helpful for many people with bipolar disorder.
References:
http://www.ncbi.nlm.nih.gov/pubmed/17547586
(a 2007 Cochrane review of mood stabilizers, showing good evidence for lithium, but also encouraging use of other mood stabilizers--which for some people could be superior to lithium-- such as valproate and atypical antipsychotics)
http://www.ncbi.nlm.nih.gov/pubmed/8120960
(a 1994 JAMA article showing the effectiveness of lithium and valproate, compared to placebo, in acute mania)
http://www.ncbi.nlm.nih.gov/pubmed/10807488
(a negative study, comparing lithium, valproate, and placebo; published in the major journal Archives of General Psychiatry in 2000--it shows very little difference between lithium, valproate, and placebo treatments with respect to relapses in bipolar patients over a 1-year period; however this study was probably biased in favour of high placebo effects and lower medication treatment effects, for a variety of reasons)
http://www.ncbi.nlm.nih.gov/pubmed/10891035
(a randomized, placebo-controlled study from Archives of General Psychiatry in 2000, showing a pronounced effect of lithium in reducing aggression in hospitalized children with conduct disorder)
http://www.ncbi.nlm.nih.gov/pubmed/16924942
http://www.ncbi.nlm.nih.gov/pubmed/3314489
(a Dutch review article from 2006, and an older article from a U.S. nephrology journal, summarizing the risk of kidney disease associated with lithium; about 15-20% of people taking lithium long-term may experience a decline in kidney function. While this decline is usually mild, I think that an alternative mood stabilizer should be strongly considered if someone is developing signs of reduced kidney function while on lithium).
But here is a start:
"Mood stabilizers" are drugs which are thought to help treat the symptoms of bipolar disorder. It is hoped that these drugs might reduce manic symptoms, prevent recurrence of manic symptoms, while also reducing or preventing symptoms of depression.
The first treatments for manic episodes were sedatives, including barbiturates and antipsychotics.
The first "mood stabilizer" though, was lithium carbonate.
Lithium itself is the third-simplest element in the universe, after hydrogen and helium. It tends to form salts. It is structurally very similar to sodium, which is a salt-forming element essential to most every life function (that is why we find sodium ions abundantly in all body fluids, and in a similar concentration in the ocean; table salt consists of sodium and chlorine atoms which join together as crystals). Yet, lithium is not normally present in the human body, and is much less common in the universe as well, compared to hydrogen, oxygen, carbon, sodium, etc.
The mechanism of lithium's action in the body, when used as a drug, is still poorly understood. Its similarity to sodium is probably essential to its mechanism. I'll add to this commentary later, but for now I will say that the mechanism of lithium salts probably involves multiple actions inside of nerve cells; these actions may modulate cellular activity.
Here are some of the clinical actions of lithium carbonate, when used as a medication in those with bipolar disorder (I will list the actions in order of how clearly proven and substantial the effect is):
1) It reduces symptoms of mania
2) It may reduce the length of a manic episode (note that just because a treatment reduces symptom severity, it may not reduce the duration of the symptoms).
3) It may prevent the recurrence of manic symptoms
4) It may prevent the recurrence of depressive symptoms
5) It may reduce depressive symptoms when they occur
There are other uses for lithium carbonate as well:
1) It can be combined with an antidepressant to improve symptom control in unipolar depression
2) It may help treat specific symptoms such as irritability and rage
3) It helps prevent cluster headaches (a type of severe, recurrent headache)
Lithium is probably most useful in "classic bipolar disorder", in which individuals experience manic episodes with elevated mood (as opposed to irritable or "dysphoric" mood), and in which the mood episodes are not recurring frequently during an average year (i.e. there is no "rapid cycling").
Here are some of the side-effects of lithium:
1) thirst, increased urine production
2) tremor (shaky hands)
3) nausea
4) sedation -- usually it is a much less pronounced type of sedation compared to antipsychotics, benzodiazepines, or other "sleeping pills". But there can be feelings of reduced energy, reduced clarity of thinking, or lethargy
5) toxicity to the kidneys -- this is not common, but needs to be checked for regularly
6) inhibition of thyroid function -- this is not permanent, nor is it harmful to the thyroid gland (in fact, it may "rest" the thyroid gland); but diminished thyroid levels, if present, requires treatment with a thyroid supplement)
7) acne or other skin rashes
8) toxicity in overdose
There have been a few studies questioning the effectiveness of lithium, particularly in terms of its value in preventing recurrent mood episodes. But for many people it does appear to be very effective, both as an acute treatment and as a preventative agent. It probably works much better as an "anti-manic" agent than an "anti-depressant".
There are various forms of lithium, and various dosing regimes. In most cases, it can be dosed simply: once at bedtime. The concentration of lithium in the blood needs to be measured periodically. Levels which are too high can increase the likelihood of toxicity (mind you, excessive levels could usually be assessed on the basis of side-effect complaints); levels which are too low may not be effective.
In my experience, some people may benefit from staying on lithium, but adjusting the dose to a point that is more tolerable for them. It may not necessarily be true that everyone needs to have a full therapeutic concentration of lithium in order for it to work. For some people, the side effects may outweigh the benefit at full doses.
However, it is a frequent situation in an emergency room, or on a mood disorders hospital ward, that people with clear histories of bipolar disorder, stable on medication, end up having a recurrence of severe mania a few weeks or months after tapering or stopping their medication (often lithium). In some of these cases, the manic symptoms may have already been building up, leading the person to discontinue their medication (rather than the other way around). But in many of these cases, it seems to me that the lithium had been protecting them, and that the recurrence of mania happened because of medication discontinuation.
There is also some evidence that sudden lithium discontinuation can provoke increased mood instability. So, while there are no overt withdrawal symptoms from stopping lithium, it should be tapered slowly if possible (I would say over 1-2 months at least).
It should be emphasized that lithium is not a perfect drug, either in terms of side effects or in terms of effectiveness. Many people on full doses of lithium still experience relapses of mania. But it is quite clear, from decades of experience, that lithium can be helpful for many people with bipolar disorder.
References:
http://www.ncbi.nlm.nih.gov/pubmed/17547586
(a 2007 Cochrane review of mood stabilizers, showing good evidence for lithium, but also encouraging use of other mood stabilizers--which for some people could be superior to lithium-- such as valproate and atypical antipsychotics)
http://www.ncbi.nlm.nih.gov/pubmed/8120960
(a 1994 JAMA article showing the effectiveness of lithium and valproate, compared to placebo, in acute mania)
http://www.ncbi.nlm.nih.gov/pubmed/10807488
(a negative study, comparing lithium, valproate, and placebo; published in the major journal Archives of General Psychiatry in 2000--it shows very little difference between lithium, valproate, and placebo treatments with respect to relapses in bipolar patients over a 1-year period; however this study was probably biased in favour of high placebo effects and lower medication treatment effects, for a variety of reasons)
http://www.ncbi.nlm.nih.gov/pubmed/10891035
(a randomized, placebo-controlled study from Archives of General Psychiatry in 2000, showing a pronounced effect of lithium in reducing aggression in hospitalized children with conduct disorder)
http://www.ncbi.nlm.nih.gov/pubmed/16924942
http://www.ncbi.nlm.nih.gov/pubmed/3314489
(a Dutch review article from 2006, and an older article from a U.S. nephrology journal, summarizing the risk of kidney disease associated with lithium; about 15-20% of people taking lithium long-term may experience a decline in kidney function. While this decline is usually mild, I think that an alternative mood stabilizer should be strongly considered if someone is developing signs of reduced kidney function while on lithium).
Noise Pollution
Peace and quiet are important for mental and physical health.
Here are a few links to references:
http://www.ncbi.nlm.nih.gov/pubmed/14757721
http://www.ncbi.nlm.nih.gov/pubmed/15936421
(A 2005 study published in Lancet which showed that noise causes increased irritability, and a negative impact on cognitive development in school-aged children)
On a related note, the use of music players such as iPods can cause permanent hearing loss, particularly if people have the volume turned up very high. People are more likely to use higher iPod volumes if the background noise level is also high. Here are some links to information and evidence:
http://www.hearinglossweb.com/Medical/Causes/nihl/mus/ipod/ipod.htm#fast
http://www.ncbi.nlm.nih.gov/pubmed/19124629
http://www.ncbi.nlm.nih.gov/pubmed/17430434
http://www.ncbi.nlm.nih.gov/pubmed/17711774
Also, the sound volume at a rock concert or a nightclub is sufficient to cause hearing damage, especially if this is an activity done regularly without hearing protection. I recommend using earplugs at rock concerts (yes, I'm serious!) Here is some evidence:
http://www.ncbi.nlm.nih.gov/pubmed/8499785
http://www.ncbi.nlm.nih.gov/pubmed/16825883
http://www.ncbi.nlm.nih.gov/pubmed/12176760
Here are a few links to references:
http://www.ncbi.nlm.nih.gov/pubmed/14757721
http://www.ncbi.nlm.nih.gov/pubmed/15936421
(A 2005 study published in Lancet which showed that noise causes increased irritability, and a negative impact on cognitive development in school-aged children)
On a related note, the use of music players such as iPods can cause permanent hearing loss, particularly if people have the volume turned up very high. People are more likely to use higher iPod volumes if the background noise level is also high. Here are some links to information and evidence:
http://www.hearinglossweb.com/Medical/Causes/nihl/mus/ipod/ipod.htm#fast
http://www.ncbi.nlm.nih.gov/pubmed/19124629
http://www.ncbi.nlm.nih.gov/pubmed/17430434
http://www.ncbi.nlm.nih.gov/pubmed/17711774
Also, the sound volume at a rock concert or a nightclub is sufficient to cause hearing damage, especially if this is an activity done regularly without hearing protection. I recommend using earplugs at rock concerts (yes, I'm serious!) Here is some evidence:
http://www.ncbi.nlm.nih.gov/pubmed/8499785
http://www.ncbi.nlm.nih.gov/pubmed/16825883
http://www.ncbi.nlm.nih.gov/pubmed/12176760
Friday, February 6, 2009
Imaginary Numbers - a metaphor
One of my favourite mathematical metaphors comes from an area called "complex analysis".
I can't resist the metaphor, because of the nature of the mathematical language involved.
I've always loved mathematics. It is enchanting, beautiful, yet infinitely challenging. There is no area within mathematics that cannot be developed into an almost impossibly esoteric branch of its own. Its theoretical abstraction must surely exceed the complexity of the physical universe (unless we consider abstract mathematical ideas to actually be part of the physical universe).
The appreciation of mathematics as an art form or as a form of esthetics has, unfortunately, been hampered by an educational approach which often leads people to experience mathematics with dread, anxiety, or despair.
Anyway, here is the mathematics:
1) A "square root" of a number is another, smaller number, which, when multiplied by itself, gives the first number. So, for example, the square root of 25 is 5--since 5 times 5 equals 25. I suppose we could add that 25 in fact has 2 square roots, since (-5) times (-5) also equals 25. This idea of a "second" square root already involves a higher degree of abstraction.
2) There are some numbers which do not seem to have any possible square root. For example, what would be the square root of (-25)? There does not seem to be any number which, when multiplied by itself, yields a negative number.
3) So, how about if we create such a number, imaginatively? Such a number has been invented, and it is called the "imaginary number", signified as "i". The imaginary number i is an abstraction, with the property that i times i equals -1.
4) What is the use of having such an imaginary number? What application could it have? Well, as it turns out, it is enormously useful in understanding and solving problems in physics and engineering. And, I think, it demonstrates a very beautiful link between phenomena that might initially seem completely different.
The exponential functions are phenomena which, if represented graphically, appear to represent rapidly accelerating growth. If something keeps doubling regularly, the growth is "exponential". Many phenomena in nature can be described using exponentials.
The trigonometric functions are phenomena which, if represented graphically, appear to represent waves, which oscillate regularly; in the case of the "sine" function, we have a "sine wave", which fluctuates, forever, between -1 and 1. Many other phenomena in nature can be described using the trigonometric functions.
There appears, at first sight, to be no obvious relationship between the exponentials, which represent unbridled growth (e.g. population growth); and the sine wave, which represents continuous, regular, well-bounded waves (e.g. the swinging of a pendulum).
But if we figure out a formula which can calculate an exponential, and a formula which can calculate a sine wave function, we find that if "imaginary numbers" are allowed, the two types of functions are variations of the same thing. Hence we have the mathematical fact:
exponential (ix) = cos(x) + i sine(x).
So here is the psychological metaphor:
The link between something which rises, escalates, explodes upwards towards infinity, and something which is stable, repetitious, and finite -- is "imagination". They are variants of the same, larger, thing, as long as you can expand your perspective of understanding.
The introduction of imagination may transform a problem of unbridled excess into one which could include stable regularity. Similarly, imagination could transform the monotony of a "sine wave" type of life into something more excitedly or wildly "exponential".
In approaching seemingly impossible life problems, I think it is important to be able to step back, and sometimes to allow an entirely new perspective or way of thinking.
I can't resist the metaphor, because of the nature of the mathematical language involved.
I've always loved mathematics. It is enchanting, beautiful, yet infinitely challenging. There is no area within mathematics that cannot be developed into an almost impossibly esoteric branch of its own. Its theoretical abstraction must surely exceed the complexity of the physical universe (unless we consider abstract mathematical ideas to actually be part of the physical universe).
The appreciation of mathematics as an art form or as a form of esthetics has, unfortunately, been hampered by an educational approach which often leads people to experience mathematics with dread, anxiety, or despair.
Anyway, here is the mathematics:
1) A "square root" of a number is another, smaller number, which, when multiplied by itself, gives the first number. So, for example, the square root of 25 is 5--since 5 times 5 equals 25. I suppose we could add that 25 in fact has 2 square roots, since (-5) times (-5) also equals 25. This idea of a "second" square root already involves a higher degree of abstraction.
2) There are some numbers which do not seem to have any possible square root. For example, what would be the square root of (-25)? There does not seem to be any number which, when multiplied by itself, yields a negative number.
3) So, how about if we create such a number, imaginatively? Such a number has been invented, and it is called the "imaginary number", signified as "i". The imaginary number i is an abstraction, with the property that i times i equals -1.
4) What is the use of having such an imaginary number? What application could it have? Well, as it turns out, it is enormously useful in understanding and solving problems in physics and engineering. And, I think, it demonstrates a very beautiful link between phenomena that might initially seem completely different.
The exponential functions are phenomena which, if represented graphically, appear to represent rapidly accelerating growth. If something keeps doubling regularly, the growth is "exponential". Many phenomena in nature can be described using exponentials.
The trigonometric functions are phenomena which, if represented graphically, appear to represent waves, which oscillate regularly; in the case of the "sine" function, we have a "sine wave", which fluctuates, forever, between -1 and 1. Many other phenomena in nature can be described using the trigonometric functions.
There appears, at first sight, to be no obvious relationship between the exponentials, which represent unbridled growth (e.g. population growth); and the sine wave, which represents continuous, regular, well-bounded waves (e.g. the swinging of a pendulum).
But if we figure out a formula which can calculate an exponential, and a formula which can calculate a sine wave function, we find that if "imaginary numbers" are allowed, the two types of functions are variations of the same thing. Hence we have the mathematical fact:
exponential (ix) = cos(x) + i sine(x).
So here is the psychological metaphor:
The link between something which rises, escalates, explodes upwards towards infinity, and something which is stable, repetitious, and finite -- is "imagination". They are variants of the same, larger, thing, as long as you can expand your perspective of understanding.
The introduction of imagination may transform a problem of unbridled excess into one which could include stable regularity. Similarly, imagination could transform the monotony of a "sine wave" type of life into something more excitedly or wildly "exponential".
In approaching seemingly impossible life problems, I think it is important to be able to step back, and sometimes to allow an entirely new perspective or way of thinking.
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