Every Nazi had Jewish ancestors. Discovering this fact alone is worth the price of Adam Rutherfords engaging and enlightening new book. A geneticist by training, Rutherford is an accomplished writer who knows how to weave a fascinating tale from scientific data as he explains that our shared ancestry is far more recent than the small group of a pan-Africa species that left the continent 70,000 years ago.

It is a popular myth that there are more people alive today than have ever died. The current global population is about 7.8 billion and increasing at the rate of 220,000 each day. It has been estimated that there have been some 108 billion members of our species, Homo sapiens. The dead may outnumber the living by almost 100 billion, but as Rutherford points out, there are more people alive as you read this than on any other day in history.

Race does existprecisely because it isasocial construct, andracism isrealbecause peopleenact it

Assuming that generations are separated by 25 years, then in every generation back through time, the number of ancestors you have doubles: two parents, four grandparents, eight great-grandparents and so on. Going back just 1,000 years generates more than a trillion ancestors 1,099,511,627,776. This staggering number is nearly 10 times greater than all of the people who have ever lived. The solution to this apparent paradox is simple: family trees coalesce and collapse in on themselves as we go back in time, with many individuals occupying multiple positions.

The last common ancestor of all people with longstanding European ancestries lived only 600 years ago, in 1400. This long lost ancestor appears on every familys tree. If you hoped for a royal connection then you wont be disappointed: as Rutherford explains, anyone alive today with a British ancestral lineage is almost certainly descended from Edward III, and all of his regal ancestors, including William the Conqueror. It may sound far-fetched, but so did six degrees of separation the idea that everyone on the planet is six, or fewer, social connections away from each other.

Remarkably, we only need to travel back 1,000 years to reach a special moment in time dubbed the genetic isopoint. Every person alive at this point in 10th-century Europe who left descendants is an ancestor of all Europeans alive today. This mind-numbing concept is a mathematical and genetic certainty that is far removed from the ancestry, family trees and identity that we learn from such TV programmes as Who Do You Think You Are?. Logically, there must also be a global isopoint, a time when the entire population of the Earth were the ancestors of everyone alive today. There is, and it was just 3,400 years ago.

How ancestry and family trees actually work shows the concept of racial purity to be pure fantasy. For humans, Rutherford explains, there are no purebloods, only mongrels enriched by the blood of multitudes. So, like the rest of us, every white supremacist and racist has African, Indian, Chinese, Native American, Middle Eastern and Indigenous Australian ancestors to name but a few.

Human genetics is the study of the similarities and differences between people and populations. Although the idea that genetic variations between traditional racial groupings have any meaningful influence on behaviour or innate abilities has been widely discredited, papers are still being published in peer-reviewed journals in which the genetics for complex human traits is sliced and diced along racial lines.

Attempts to justify racism have long been rooted in science, more accurately pseudoscience. Rutherford understands that racism is a social phenomenon, but rightly believes that when science is warped, misrepresented or abused to justify hatred and prejudice it must be challenged. He focuses on what genetics says about skin colour, ancestry, intelligence, sporting prowess, and about so-called racial purity and superiority. And he attempts to equip the reader with the scientific tools necessary to tackle questions concerning race, genes and ancestry, as he explains what DNA does and does not reveal about the concept of race.

No one has ever agreed how many races there are, nor what their essential features might be. The emergence of the pseudoscientific approach to human taxonomy that relies on physical traits such as skin colour or physiognomy coincided with the empire building of European powers. Unsurprisingly, the invention of race occurred in an era of exploration, exploitation and plunder.

Skin colour may be the most obvious difference between people but it has little to do with the total amount of similarity or difference between individuals and between populations. If we accept that people are born with different innate capabilities and potential, then how these abilities cluster within and between populations has more to do with history and culture than DNA and biology. Studies reveal that genetic differences between populations do not account for differences in academic, intellectual, musical or sporting performance between those populations.

So-called racial differences are literally just skin deep: genetics and human evolutionary history do not support the traditional or colloquial concepts of race. As a result, Rutherford argues, we are prone to say race doesnt exist, or race is just a social construct. However, race does exist precisely because it is a social construct, and racism is real because people enact it. One has to admire his desire to challenge Jonathan Swifts dictum: Reasoning will never make a Man correct an ill Opinion, which by Reasoning he never acquired.

How to Argue with a Racist: History, Science, Race and Reality by Adam Rutherford is published by Weidenfeld & Nicolson (RRP 12.99). To order a copy go to guardianbookshop.com. Free UK p&p over 15.

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