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Mathematics for Electronics 1

Number Systems

The numbers in the system you are most familiar with are called Hindu-Arabic Numerals, invented by Indians, and brought to Europe by Arabs. Europe was using the Roman Numerals at that time, a system that was probably invented to keep track of soldiers in military units and divisions. It probably worked well, looking at the military successes of romans. For example if they had military units of X (10) soldiers, and if a soldier left the group, they would put a line on the left to show the reduced number IX (9). If they had another group with one extra soldier XI (11), they could put these two groups together to form two proper units of X soldiers each. I’m making this up, I don’t know if they really used this number system that way. I can’t see any other advantages of this system, because it was really cumbersome to do trade or scientific/mathematical calculations using this system. They used to have specially trained people in markets to do the calculations for everyone, because they were quite difficult. When Arabs started to trade in these markets, they were already using the Hindu Numerals and were a lot more efficient and a lot faster at doing calculations compared to the Europeans. The secret was a place based number system, and a mysterious new number from India they called Siffer. This really baffled the Europeans, they didn’t trust the new number system and the calculations, even though they did give the correct answers fast. They were so confused by this Siffer, that they still use the word Cypher to mean something coded in a way that it could not be read and understood easily. For their understanding, they used to remove the Siffer, or Decypher the Hindu numbers. These days, the whole world uses that number system, which many classified as dark arts in the past.

This Hindu-Arabic number system is based on 10 Numerals (or Symbols), i.e. 0 to 9. What most people don’t realise is that it’s not the Decimal nature of the system that has made it successful. It’s actually the Cypher, or Zero, giving value according to the place of a Numeral in a group of Numerals, and uniform multiplicative interval between all adjacent numeral positions. In fact, most of the other number systems currently in use are also based on these three properties of the Hindu number system, i.e. they all use Zero, and they all are based on the positional value of the numerals, and they all have uniform multiplicative intervals. For example the binary, octal, and hexadecimal systems are all based on the same principles. Without these number systems, it is highly unlikely that electronic computers or any digital devices would have been invented by now.

Positional Value of Numerals, and the Use of Zero to Achieve That

Let’s consider 3 numeral or 3 digit numbers:

A) 001 (One in decimal or One in binary)
B) 010 (Ten in decimal or Two in binary)
C) 100 (Hundred in decimal or Four in binary)

As we move the position of 1 from right to left, they value of 1 increases to ten and then a hundred. Where we write a number give it a corresponding value. This was not true for roman numbers for example:

A) IX (Nine)
B) X (Ten)
C) XI (Eleven)

Here, the position of X doesn’t have positional value. This makes it harder to do calculations. For example if we were to add the three numbers in both systems:

0 0 1 IX
0 1 0 X
+1 0 0 +XI
1 1 1 ?!

Well this one is simple because IX + XI = X + X as explained earlier, and X + X + X = XXX (Thirty). But as the numbers grow larger, it becomes really difficult. And if you have to do multiplication, division, and find roots… good luck! While these mathematical operations are quite easy in the Hindu number system.
In decimal system, one + ten + hundred = one hundred and eleven, or 111, and in binary system, one + two + four = seven or 111 binary. We added the same way in both the decimal and the binary system and got correct answers.

Forget about calculations, just simple reading of roman numbers can become quite complex for us these days, for example:

1989 = MCMLXXXIX
2022 = MMXXII

You may find these numbers in movie titles, showing the year of their release, which must appear as a Cypher to most people. Apart from that, we use roman numbers as page numbers and as document parts numbering, provided they don’t go too high.

Multiplicative Interval Between Adjacent Numerals

Units

In order to measure and calculate quantities of different types, we have devised systems of standardised measures. Majority of these units have been taken arbitrarily, at different times and at different places. For example Foot is a unit of length. A metre is also a unit of length. Neither is better than the other; there are many other units of length and we can use any of them.

Constants

A value, or a condition, which does not change is called a Constant.

Physical Constants

These are certain quantities that we have determined to be constant

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