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Written by Bryce Ringwood

The Dawn of Arithmetic

Arithmetic in  the form of counting seems to have been around for about 35 000 years (give or take). It seems that early hunters would make notches on pieces of bone for each animal they killed in the hunt. As civilisations rose and fell in succeeding millenia, mathematical thought developed in response to the needs of the people. Mathematics were employed by surveyors when building the great pyramids. One of man's earliest obsessions has been with the skies and many civilisations had a priesthood who used mathematics to predict the seasons and astronomical events.

With the advent of trade, some means had to be devised to manage the value of transactions. We call it money, but before money, a record of the number of tiger skins swapped for a given number of beads had to be made. This gave rise to the first calculators - around 2700BC.  They would have consisted of clay tablets with grooves containing 5 stones. (The ancient greek word for stone is "calculus" - from which we get the word calculator). Later, a portable version consisting of beads strung on wires within a wooden frame called an abacus came into use. This device has persisted until the present day.

Mathematics is more than mere counting, however, and it is the ancient Greeks who elevated it to a system of pure thought. Greek thought in the period 600BC was centered around harmony and the relationships of things to each other. They introspected about prime numbers, rational numbers and so on. Pythagoras is credited with the proof of his famous theorem which we remember as an algebraic formula although algebra was unknown to the Greeks at that time. Indeed, the Greeks represented their abstract mathematical thought as geometric elements until some time in the 3rd century AD. At this time the greek Diophantus used algebraic symbols in his mathematical proofs. As with many great discoveries and inventions, we can't truly say Diophantus was the father of algebra, because there are other great mathematicians who can lay claim to have been using algebra at the same time.

You may be surprised that up until 1200 AD everyone got along without the "number" zero. If you look at Roman numerals - you will see there is no zero. If we count to 10 in Roman numerals :  I, II, III, IV, V, VI, VII, VIII, IX, X. - there is not a "zero" in sight. Thus "10" has a symbol all to itself. It was not until 1200AD that zero  was used in the Western world. Credit goes to Fibonacci - the creator of the series that bears his name.

From this period on, we see mathematics awakening into the modern era. What we are interested in is the way we can apply the subject to further our understanding of the topics on this web-site, where I have regrettably had to introduce formulae.

Calculators

Human minds are not computers, nor do they resemble computers in any imaginable way. Most of us are pretty bad at doing calculations in our heads, which is why calculators were invented. In 1642, Blaise Pascal devised the first mechanical calculator. It was capable of addition and subtraction. Later machines could multiply by repeated addition and divide by a process of repeated subtraction and remaindering. Examples of these mechanical horrors were very much in evidence when I started my career. Many engineers will remember the FACIT and Brunsviga calculators and the joys of unjamming them.

These were good for some engineering and surveying calculations, but the commonest device by far was the slide rule. Whilst many people are adept at addition and subtraction, the operations of division and multiplication are slow and tedious. Slide rules made such calculations easy. In addition, they provided trigonometrical functions.

In the early 1970s, Busicom introduced the first pocket electronic calculator (launching Intel on its way with the 4004 chip). In 1972, Hewlett Packard introduced the HP35 scientific calculator, which was popular in many design offices. To commemorate its debut, HP  marketed the HP-35s in 2007 to commemorate the 35th anniversary of the HP35. They are a tad pricey  - but don't worry, here's your very own HP-35 calculator This simulator was written by Neil Fraser in Javascript. You can download your own copy from the HP museum web site. Alas, it can't simulate the really nice feel of the HP-35's keys.

I was quite surprised to find such a wide selection of calculators on sale at my local stationery. Most mobile phones have a calculator which is perfectly adequate, and downloads of scientific calculators are readily available at a nominal price. Still, new models are coming out all the time, with Texas Instruments "TI-84 silver" having a colour screen.

Operators (and Operands)

The arithmetic operators are

{\displaystyle \times}
{\displaystyle \div}
{\displaystyle +}
{\displaystyle -}

We do not have an operator for "raising something to the power of" or exponentiation, for example {\displaystyle 2^3} .The BASIC computer language uses the "^" and FORTRAN ( a BASIC - like scientific programming language) uses "**".

Exponents
{\textstyle 2^3} Means 2 X 2 X 2. It is not obvious what {\textstyle 2^{0.7}} might mean.

Later on, I will attempt to explain. For now, use a calculator to verify the following:

{\textstyle 2^3 = 8}
{\textstyle 2^0  = 1}
{\textstyle 2^{(-1)}  = 0.5}
{\textstyle 2^{0.5} = \sqrt {2}}

Now observe that (for example) {\textstyle4^4 = 4^2 \times 4^2=4^1 \times 4^1 \times 4^1 \times 4^1 = 4^1 \times 4^3}. All the exponents add up to the same amount - namely 4. We can generalise this to negative and fractional exponents thus:

{\displaystyle {4^{0.5}  \times 4^{0.5} = 4^1 = 4}}
and
{\textstyle {8^{1/3}  \times 8^{1/3} \times 8^{1/3} = 8^1 = 8}}

In general any number to the power of 1 divided by  something gives you the somethingth root  i.e.{\textstyle {somethingth \root \of (number)}}

Now - how about negative exponents ? - Well {\textstyle {8^{-1}  \times 8^{1} = 8^0 = 1}} I will let you do the calculations for other negative exponents. Clearly, an exponent of -1 provides the reciprocal (1 over) the number.

Finally - fractions. Now we can see that :

{\displaystyle {2^{0.7} = \left( {{^{10}}\root \of 2} \right) ^7} }
You might want to verify this with a calculator.

{\textstyle (-8)^{(2/3)} }
The answer is 4 - Traycalc, and my Sharp calculator both give an error message. The cube root of -8 is -2, and -2 squared is 4 right ?

Well, since I wrote traycalc, I know that it summarily dismisses any negative number with a fractional exponent. It looks like I'm not alone when it comes to possibly faulty calculators. We will see that the HP-35 also can't work this out.

Traycalc gives the error "fractional exponent of negative argument". Quite. The answer is, I think, that calculators are not designed to perform operations on complex numbers - which we will look at in another article. In general, calculators will not give you an answer for the  square root of minus one. Also, you will note that the square root of one has two answers, namely 1 and -1. Which answer should the calculator give? It turns out that the cube root of 1 has three answers - and so on, but we'll have to wait until a later article before we continue this discussion. For those who are impatient - here's a clue.

Doing Arithmetic Calculations

The problem with engineering calculations is their apparent complexity. For example:

{\displaystyle {5.0 \times { (1 - 2.71828^{-7.0 \div {5 \times 10^5 \times 20.0 \times 10^{-6}}}) } }}

This is from the capacitor article and its a mess.

There is a simple mnemonic, which is intended to remind us of the standard way of doing things - BEDMAS. This means Brackets,Exponents,Multiplication,Addition,Subtraction. If we obey the rule, we get:

{\displaystyle {5.0 \times { (1 - 2.71828^{-7.0 \div {10}}) } }}

Working out the exponent first (and cheating by putting brackets round the -.7 ) - we get the (intended ?) answer 0.511. Unfortunately the BEDMAS rule doesn't give us much idea of where in the order of things the unary + and - operators have got to go.

For amusement, you might like to calculate the sum

{\displaystyle{3 + 5 \times 7}}

on a number of pocket calculators. My mobile phone and Microsofts calculator give the answer as 56. The traycalc gives the correct answer of 38. BEDMAS says you should multiply 5 times 7 and then add 3.

You will have gathered by now that BEDMAS is a convention not a law. Some programming languages and calculators do not obey the convention - so brackets have to be used to sort things out.

Let's rewrite that wretched bit of arithmetic:

{\displaystyle {5.0 \times { (1 - 2.71828^{(-7.0 \div {(5 \times (10^5)) \times (20.0 \times (10^{(-6)}))})}) } }}

{\source{Hey! - That's all very well but how do I cope with all those brackets?

Hewlett Packard's HP-35 Calculator and RPN

If you are used to a "conventional calculator" - you might be confused at the sight of the HP-35 a calculator that had no equal. Well, no equals symbol,anyway.

If Jan Lukasiewicz was the inventor of "Polish notation", HP were the adopters of Reverse Polish Notation" or RPN. In RPN, the operators are placed after the operands, so our little sum is performed on the HP-35 by the following keystrokes:

 5 Enter 7 X 3 +

In their manual, HP recommend that you work out the things in the brackets first, and go to great lengths to explain how it is much less likely that you will make a mistake using RPN than if you use a conventional calculator.

You might like to calculate the value of our horrible example, after some practice with simple calculations.

Call me a party pooper, but I think the real reason HP used RPN is that it is computationally efficient. RPN calculator programs would fit more easily into the limited memory and address space of those early calculator chips.

Pratice this (Preferably on more than one calculator):

{\textstyle{9 + 5 \times 2}}
{\textstyle{(9 + 5) \times 2}}
{\textstyle{9 + 5 ^{2 \times 3.5}}}
{\textstyle{9 + 5 ^ 2 \times 3.5} }
{\textstyle{ 50 \times { \sqrt{1 - {{40^2 }\over {300 ^2}}}}}}

Percentages

In 1986, calculators still represented an estimated 41% of the world's general-purpose hardware capacity to compute information. This diminished to less than 0.05% by 2007. - "The World’s Technological Capacity to Store, Communicate, and Compute Information".

a) Do you believe that ? and

b) Do you understand it ?

The percent symbol "%" is just a wierd way of writing "divided by 100". I guess that's what the two little zeroes and the slant line are for.  So 14% is just another way of writing "0.14"But, 14% VAT looks better than "0.14" VAT.

Its just a way of picturing things.

Oh - and that intro to this section - I'll believe it when my colleague stops asking to borrow my calculator.

Factorials

Designated with a splash "!" symbol. Example 4! = 4x3x2x1; 7! = 7x6x5x4x3x2x1 etc. Traycalc has a FAC() function to evaluate them. I don't recall other calculators having this built-in.

Programmable Calculators

The "programmable calculators" of the early 1970s were rather awful. Thankfully, Hewlett-Packard introduced the HP-65 in 1973(?). Anyone who could use a calculator could program it. All you had to do was jot down the keystrokes you were going to use and enter them into the HP-65's memory in program mode. If the program worked,you could record it on a small magnetic card. At that time, the alternative was to write a FORTRAN program on a deck of punched cards .. then Wait to see if it worked. If someone dropped your cards, or they got shuffled, it was very bad news....

HP made quite a number of programmable calculators from then on. Alas my 2nd hand HP9825 and HP-87 both died on me. I was particularly sorry to see the HP-87 go - it was programmable in BASIC. The only programmable I have at present is a SHARP PCE 500 - also programmable in BASIC. It is euphemistically referred to as a "Pocket Computer" - but you'll need very large pockets.

In 1981, the first IBM PC had ROM BASIC - making it a programmable calculator by some definitions. GW-BASIC seems to have been abandoned by Microsoft. It is almost the same as the IBMs ROM BASIC. Still, its not the same as keystroke programming. We will look at automating calculations in the section on formulae.