In 1942, the unthinkable happened. This “help wanted” ad appeared in the Philadelphia Evening Bulletin: “Looking for Women Math Majors.”

The ad was placed by the US army, which was hiring women to work at the Moore School of Electrical Engineering, at the University of Pennsylvania. Kathleen McNulty was only 21, a brand new graduate of Chestnut Hill College, but she knew nothing like this had ever appeared outside the “Male Help Wanted” section of any newspaper before the US entered the second world war.

The war was transformative for Black Americans, who were finally integrated into the army after the war ended, and gay Americans, who discovered for the first time that there were thousands of others with the same secret desires. But no one benefited more from the war than women, whose career opportunities exploded as millions of men left farms and factories to fight the Nazis and Japan.

Two weeks after she graduated, McNulty responded to that Evening Bulletin ad. She was immediately hired by the army. A couple of years later she became one of six women who would program the first modern computer. Their stories and the saga of that computer’s invention are the subjects of this beguiling book.

The author, Kathy Kleiman, now a law professor at American University, was a computer programmer in high school. As an undergraduate at Harvard, she discovered two photos with women standing in front of the Eniac, the 8ft-tall and 80ft-long behemoth invented for the army by J Presper Eckert and John Mauchly. From that moment on, Kleiman became obsessed with learning the identities of all the earliest women programers.

The result of that magnificent obsession was a documentary in 2014 and this book, which melds social history with the major events of the second world war and the biographies of these six remarkable pioneers to produce an irresistible narrative.

The others besides McNulty were Jean Jennings Bartik, Frances Elizabeth Snyder, Frances Bilas Spence, Marlyn Wescoff Meltzer and Ruth Lichterman Teitelbaum. Among them were a Catholic, two Jews, a Quaker and a Presbyterian.

They were all hired for the army’s Philadelphia Computing Section and their title was assistant computer, which meant they did lengthy calculations on old-fashioned mechanical machines. All started with the rank of “subprofessional” or “subscientific” just because they were women, but their starting salary of $1,620 (about $27,000 now) was double that of any secretary.

Like all successful pioneers from previously discriminated-against groups, each of the women had to be exceptional to succeed. Marlyn Meltzer, for example, quickly became famous for never making a mistake in any of her calculations.

They had to overcome every traditional sexist hurdle, including an “overly familiar doctor” who performed the entrance physical on Jean Bartik and invited her to his home to complete it.

“The old farm boys had taught me well to stay out of secluded places such as haylofts,” Bartik remembered, so she refused to go the doctor’s home. Remarkably, when she reported “what kind of lecher he was”, the army stopped using him for physicals.

How Eckert and Mauchly convinced the army to finance the world’s first all electronic programmable computer is the story which animates a long section of Kleiman’s book.

The Electronic Numerical Integrator and Computer (Eniac) originally had a single goal: to improve the accuracy of American artillery. Early in the war the army figured out that it had to account for distance, humidity, air density, temperature and the weight of the shell. When troops took artillery units to the desert, the difference in soil from Europe required a whole new set of calculations.

Before the new computer was invented, the women who would program it had to use desktop calculators. They were “essentially pushing the missile’s motion forward across its arc in the sky, step-by-step, to its explosive end at the completion of its journey”. A precursor of the Eniac had dozens of motors, thousands of relays, 2,000 vacuum tubes and 200 miles of wire – “all to solve just one ballistics trajectory”.

The Eniac had an astonishing 18,000 vacuum tubes, and the failure of any one of them could ruin its calculations. One of the inventors’ eureka moments occurred when they realized they could make the tubes more reliable by under-powering them.

One of the many astonishing accomplishments of these pioneering women was the system they developed to identify the location of any defective tube in the huge machine. And because nothing like it had ever existed before, the only way they could teach themselves to program it was by studying its blueprints.

“They gave us these great big block diagrams … and we were supposed to study them and figure out how to program it … Well, obviously we had no idea what we were doing,” one remembered.

But Marlyn Meltzer “had a sense they would grope it out and figure it out together”.

Incredibly, she was right. But because most male-written histories of this incredible invention omitted the crucial role of these women, this book marks the first time they have all received the gigantic credit they deserve.

• Proving Ground: The Untold Story of the Six Women Who Programmed the World’s First Modern Computer is published in the US by Grand Central Publishing