25 December 2013

Large numbers in and around us

By Dr Manas Kumar Haldar

“Ancient Indians were afraid of many things but never afraid of large numbers. So they conjured up large numbers.” I was talking to my nephew’s 10-year-old son Arjun. “Do their large numbers exist in and around us?” he asked. “Bring your calculator; I’ll show you some large numbers,” I said. Arjun brought his calculator (a Casio FX-350 MS) and complained that it sometimes gives funny answers. “See, if I multiply 102 by 102, it gives me the correct answer 10404. Now if I put three more zeroes inside the numbers, 100002 times 100002 should give me 10000400004 but the calculator shows 1.00004 x 1010.” “Well, the calculator has limited display, so it does its best. 1010 is one followed by 10 zeros. The upper 10 is called power.” “But if I multiply 1.00004 by 1010, I get 10000400000” said Arjun. “That error is small for a large number like this,” I said. “So for estimates of large numbers we can use powers of 10.” “Can power be negative?” asked Arjun. “Yes, large negative powers represent very small numbers. For example 10 to the power –N represents a number with N-1 zeros between the decimal point and one.”

Then I explained to Arjun that 10 to the power N multiplied by 10 to the power M is 10 to the power N+M and 10 to the power N divided by 10 to the power M is 10 to the power N-M. “My kid sister’s age is one. What is her age in power of 10?” asked the naughty boy. I answered “Ten to the power zero because 10 to the power N divided by 10 to the power N is one, but it is also 10 to the power N-N or 10 to the power zero.”

Arjun continued pestering me about large Indian numbers in the real world. “Well consider the distance between the sun and earth. It is about 10 to the power eight kilometres. Now consider the distance of the sun from the centre of our Milky Way galaxy. It is about 10 to the power 17 kilometres.” Arjun interrupted, “Can’t you be down to earth?” “Well, consider the number of gas molecules in the atmosphere …” Arjun was thoroughly bored. “You learned electronic engineer,” he said. “How many electrons do I have in my body? Dad says that electrons are the smallest particles.” My nephew intervened, “Arjun, don’t sneer at your grandpa and give him a break. He is here on holiday.”

I laughed. “If you want an accurate answer Arjun, I have to find which molecules your body has and how many of each type there are. Then knowing the number of electrons in each molecule, I can calculate the number of electrons in your body.” “Anything easier?” Arjun asked. “OK, tell me your weight.” “It’s 34 kilograms,” answered Arjun.

“Listen carefully. Your body is made of molecules and molecules are made of atoms. So your body is made of atoms. An atom consists of protons, neutrons and electrons. The weight of a proton/neutron is about two thousand times the weight of an electron. So the weight of an atom and hence the weight of your body is approximately the weight of its protons and neutrons. The weight of a proton/neutron is about 10-27 kilograms. So how many protons and neutrons does your body have?” “Easy”, Arjun answered, “I don’t need the calculator now. Write 34 kilograms as 3.4X10 like what the calculator showed. Then considering only powers of 10, the number is about 101 divided by 10-27 or about 10 to the power 1-(-27) or 10 to the power 28.”

I continued “Your body has many elements but most of them are hydrogen, carbon, oxygen and nitrogen. These elements usually have only protons and electrons but they may also have neutrons. Those that have neutrons are called isotopes. For example, hydrogen has one proton and one electron; but one of its isotopes, deuterium has one proton, one neutron and one electron. Isotopes are rare. So ignoring neutrons, the number of protons in your body is about 10 to the power 28.”

“Now we can find the number of electrons. The number of electrons in an atom is always equal to the number of protons. So your body has about 10 to the power 28 electrons. This is one followed by 28 zeros. This number was called one ‘nahuta’ by ancient Indians.”

“Dad, I have become a scientist,” an exasperated Arjun cried out. “Not quite,” said my nephew. “Finish your homework first.”

Dr Manas Kumar Haldar is an Associate Professor with the Faculty of Engineering, Computing and Science at Swinburne University of Technology Sarawak Campus. Readers with a simpler and more accurate estimate of the number of electrons in Arjun’s body may contact him at mhaldar@swinburne.edu.my