If you send a cheek swab and $220 to a company called Oxford Ancestors in England, you will receive a certificate -- "suitable for framing," according to the company's Web site -- documenting whether you are descended from one of seven European "Daughters of Eve" or 29 other "clan mothers" who lived many thousands of years ago. For $319, a firm called DNAPrint genomics Inc. in Sarasota, Fla., will tell you what proportions of your ancestors are from the Americas, Europe, Africa and eastern Asia. Family Tree DNA in Houston, for prices ranging from $149 to $499, will allow you to find "genetic cousins" by comparing your DNA to the genetic sequences in the company's databases.

These and other genetic tests now being rushed to market may seem a harmless diversion for people eager to uncover their roots and cultivate an ethnic identity. But they have the potential to do great mischief. By focusing on some genetic connections and not others, they reinforce the popular misconception that each of us is descended from a relatively small and distinct number of ancestors. They ignore the dense web of genetic and genealogical connections that bind all human beings into a single, interconnected species.

Ancestral pride would not be such a dangerous thing were it not linked so closely to issues of race and ethnicity. (DNAPrint genomics, for example, suggests that one use for its test is "to validate your eligibility for race-based college admissions or government entitlements.") Some Americans may think of themselves as having ancestors only from England, but that means they probably also think of themselves as purely European or "white." Similarly, most African Americans might think that most or all of their ancestors are from Africa, despite the long history of genetic mixing in the Americas.

In fact, all human beings have ancestors from throughout the world. The reason we think about our ancestry in such limited terms is because we are considering just our immediate ancestors -- our two parents, four grandparents, eight great-grandparents and so on. We are overlooking the more distant past, where the power of exponential growth kicks in.

Math teachers sometimes introduce their students to exponential growth through the story of Sessa, the 15th-century Indian mathematician who supposedly invented chess. According to legend, the king was so delighted with the game that he told Sessa to name his reward. Sessa said, "Sire, I would like to have two grains of wheat for the first square of the chessboard, four grains of wheat for the second square, eight for the third, 16 for the fourth, and so on until the board is filled." The king was greatly annoyed at being asked for such a trifling award. "You have insulted me by not asking for more," he told Sessa. "Go, and my servants shall bring you your sack of grain."

At dinner the king asked if Sessa had been paid and was told that the court mathematicians were still calculating the exact amount of grain Sessa should receive. The king frowned; he'd expected that such a simple task would be carried out more quickly. He said, "Before I am awake tomorrow morning I want Sessa to have his reward." The next morning the king called for his chief mathematician and asked how many grains of wheat Sessa had received. "Sire," said the mathematician, "it is more than the amount of wheat that exists anywhere in the world. The number of grains that must be placed on the final square of the chessboard is 18,446,744,073,709,551,616. Family trees work just like Sessa's reward. Say you wanted to make a family tree showing all of your ancestors 64 generations ago, which is about 16 centuries ago, around the time of the fall of Rome. The number of your ancestors in that generation would be 2 to the 64th power -- the same as the number of grains of wheat on the 64th square of Sessa's chessboard.

But how can that be? Eighteen quintillion people are more than ever have lived or ever will live. The way out of the puzzle is to realize that over time, distant cousins, or sometimes not-so-distant cousins, marry and have children. Every time that happens, the same person occupies two or more of the positions on the family tree of those children. If you actually could construct your family tree for 64 generations, some names would show up in that earliest generation many trillions of times.

Still, the general point remains. We think we have relatively few ancestors because our historical vision is so shallow. Just a few centuries into the past, the number of our ancestors becomes immense.

A particularly important transition occurs between about 10 and 25 generations ago. Going back 10 generations, each of us has about 1,000 ancestors (ignoring, for a moment, the effects of cousin marriages). Look back 25 generations and each of us has more than 30 million. At that point, the number of our ancestors becomes comparable to the entire populations of the parts of the world where our ancestors lived. For example, someone of European heritage could include on his or her family tree the majority of the 30 million to 40 million adults who lived in Europe in about the year 1400. And once your ancestors include most of the people from a particular part of the world, that situation continues indefinitely back into the past. Therefore, everyone of European ancestry today is descended from most of the people who lived in Europe before 1400.

That's the problem with the Seven Daughters of Eve business, an application of gene testing that makes many geneticists cringe with embarrassment. The test analyzes a small piece of genetic material in our cells known as mitochondrial DNA. Children inherit this tiny loop of genes only from their mothers, who in turn got it from their mothers, and so on back through time. A test of mitochondrial DNA therefore samples a person's maternal lineage.

But this represents just a single twig from an extremely bushy family tree. Of the millions of people who were your ancestors several centuries ago, just one gave you your mitochondrial DNA. This radical narrowing of genetic lineages is essentially why most Europeans have mitochondrial DNA from one of just seven women who lived between 10,000 and 45,000 years ago. But in fact all Europeans are descended from all seven of these women. (For that matter, so is everyone else in the world, because migration between continents causes networks of ancestry to interconnect.) That a given European happened to get mitochondrial DNA from one of them is more a matter of random chance than anything else.

There is one way that a person could have only English ancestors for a considerable time into the past. Say that England were settled by the first modern humans to enter Europe from Africa about 40,000 years ago, and that thenceforth a genetic blockade were thrown up around the island, so that no new immigrants to England were allowed to marry into the population. In that case, Americans of English ancestry could have ancestors who lived only in England -- at least for the past 40,000 years (before that, their ancestors would be African).

But the real world doesn't work that way. Throughout history, people have been moving from village to village, from region to region, and from continent to continent. Even isolated Pacific islands are continually absorbing immigrants. Over just the past 3 millennia, Celts, Angles, Saxons, Jutes, Vikings, Normans and other peoples have moved to England and married into the population.

The constant movement of people from one part of the world to another greatly complicates the task of saying exactly where your ancestors lived. For example, DNAPrint genomics says that its test can determine the proportions of your ancestors who were living in different parts of the world before the Age of Exploration, around 1500. But Africans have been moving in substantial numbers to Europe for more than a millennium, so pieces of DNA that originated in Africa could have taken a lengthy detour through Europe (and vice versa).

Such tests harbor many other uncertainties. Most of the common genetic variants are found in equal proportions in all human groups, because we are a relatively young species and have mixed extensively throughout history. So tests of ancestry have to look at the rare variants that have different frequencies among groups. These tests therefore are not finding pieces of DNA that are distinctly African or European. They are producing statistical estimates of ancestry.

For that reason, error rates are relatively high. DNAPrint genomics says that if tests show that fewer than 5 percent of your ancestors come from a particular continent, the results should be taken "with a grain of salt." But at that level of uncertainty, the test could miss an African ancestor in an otherwise European lineage who lived just five generations ago.

The ultimate question that must be asked of any genetic test is, "Why is it important for me to know this information?" Granted, these tests may be able to verify family stories of exotic ancestors, whether those ancestors are European, African or Native American. Then again, they may show that the man listed on your birth certificate is not your real biological father -- caveat emptor.

The relentless focus on differences also obscures our remarkable genetic unity. Geneticists have demonstrated that we are all descended from a small group of people, perhaps just a few thousand, who lived in eastern Africa about 6,000 generations ago. So if everyone in the world were to take their family trees back 6,000 generations, all of the names on those trees would be the same.

In the time since then, some people left Africa, and groups of their descendants in different regions acquired distinctive facial features and skin colors. Yet all humans are too closely related to have developed fundamental biological differences. The last group of humans who had such differences -- the Neanderthals -- died out about 30,000 years ago.

If asked about my own ancestry, here's what I would say: For the past several generations, all of my ancestors have been Americans. Before that, most of them, though certainly not all, were Europeans. And before that, as is the case for everyone else in the world, all of my ancestors were Africans. I guess I could pay a genetic testing company to tell me as much. Personally, I'm going to hold on to my money.

Steve Olson, who lives in Bethesda, is the author of "Mapping Human History: Discovering the Past Through Our Genes" (Houghton Mifflin). His book was one of five finalists for the 2002 National Book Award for nonfiction.

Among the tools to discover our ancestors are DNA testing kits (above), genealogical maps (left) and, according to the author, a dose of common sense.