﻿ Magic Schoolbook: The Education System for the Web

# How to Use a Scientific Calculator

0 _______________________ _______________________ ____ ____ || || | |

exp

ac

ce

Pi

0

(...)

.

inv

° ' "

drg

Deg

sin

cos

tan

( ` / _ x )

squareroot

(x²)

roots

powers

ln

log

1/x

x!

WARNING: this calculator's newly written and may still contain bugs. (Sep 2010)

ME

MR

0 _______________________ _______________________ || ||

### Keyboard entry:-

Number keys work as you'd expect, but "o" also works as zero.

For · use "p" (or "." if it doesn't make the browser start a search).

For = use "=" or "e".

For AC use "w" (wipe), for CE use "u" (undo).

For + use "a" (or "+" if it doesn't zoom the screen).

For - use "m" (or "-" if it doesn't zoom the screen).

For × use "x" or "*".

For ÷ use "d" (or "/" if it doesn't make the browser start a search).

For ° ' " use "h" (hour), for x°y'z" use "g" (which is next to "h").

For sin use "s", for cos use "c", for tan use "t", for inv use "i".

For x^2 use "q" (quare). For square root use "r".

For ^ use "^". For inv ^ use "v".

For log use "l". For 10^x use "k" (which is next to "l").

For ln use "n". For e^x use "b" (which is next to "n").

For 1/x use "z" (because it has a diagonal line in it and we're running out of keys).

For ! use "!".

For M in use "f" (file in) followed by a letter. For MR use capital letters.

Keys "y" and "j" not yet assigned.

All the content below can be reached by clicking on links at the top - it is not designed to be read by scrolling down through the whole page.

### All Clear

The AC button clears the calculator ready for the start of a new calculation, though it doesn't clear the memory or memories.

### Clear Entry

The CE button clears the number most recently entered on the calculator, but without clearing anything else, so if you type a number in wrongly you can clear that from the screen and enter it correctly without having to retype any earlier parts of the calculation leading up to that point. Usually it deletes the whole number, but an alternative is for it to work like a delete key so that you can delete the number digit by digit.

### Brackets

On simple calculators when you type in 2+3×4= it will add the 2 to the 3 first to get 5, and then it will multiply the 5 by 4 to get 20, but more serious calculators hold back the addition and do the 3x4 part first before adding the 2 on at the end. This is because there is a rule that × and ÷ are higher priority operations than + and −. Powers are higher priority still, so if you type in 2+3×4^5= the 4^5 bit is processed first to get 1024, then the 3×1024 is done (3072), and then the 2 is added to that (3074). Sometimes in the middle of a calculation things become complicated as you may want to force an addition to be done before a multiplication, so you can isolate the addition by putting it in brackets which will cause the calculation on the content between the brackets to be calculated as soon as the closing bracket button is pressed.

An example of this is 2×3×(423-376)÷27=. In this example, the 2 and 3 are multiplied first, then the subtraction is done before the second multiplication, and finally the division is done. If there were no brackets, the 2 and 3 and 423 would be multiplied together, then the 376 would be divided by 27, and then the second answer would be taken away from the first. It's possible to use brackets within brackets for deeper and deeper complications, though it's usually easier just to calculate things in parts and store the answers to those parts in memories before combining them. Anyway, the function is there if you need it, and on this particular calculator it can handle millions of levels. I have also programmed it in such a way that you can type the opening bracket after the first number to go in the brackets because frequently you don't think of pressing the bracket key in time and have to start from scratch, but here you can simply press it after the event and keep going, just so long as you haven't typed an instruction key such as +/−/×/÷ before it as well.

### Exponent

Really huge numbers and really tiny fractions can only be shown on calculator screens by writing them in a differnt way, so look at this series of numbers in which each number is ten times bigger than the one above it (the blue numbers are the ones which can be displayed normally on most calculators, but for bigger or smaller numbers they usually have to be displayed as shown in red here):-

1×10^-9 = 1.-9
1×10^-8 = 1.-8
1×10^-7 = 1.-7
1×10^-6 = 1.-6 = 0.000001
1×10^-5 = 1.-5 = 0.00001
1×10^-4 = 1.-4 = 0.0001
1×10^-3 = 1.-3 = 0.001
1×10^-2 = 1.-2 = 0.01
1×10^-1 = 1.-1 = 0.1
1×10^0 = 1. 0 = 1
1×10^1 = 1. 1 = 10
1×10^2 = 1. 2 = 100
1×10^3 = 1. 3 = 1000
1×10^4 = 1. 4 = 10000
1×10^5 = 1. 5 = 100000
1×10^6 = 1. 6 = 1000000
1×10^7 = 1. 7 = 10000000
1×10^8 = 1. 8
1×10^9 = 1. 9
1×10^10 = 1. 10

The green numbers show what the red numbers actually mean, the symbol ^ meaning that the 10 before it has to be multiplied by itself as many times as stated by the number after the ^. If the number after the ^ is negative, the 10 has to be divided by itself one more time than the number after the ^-. Calculators on computers may put a letter "e" in the gap and they may also put in a "+" sign if the exponent is not negative, so 1.-4 may look like 1e-4 and 1. 4 may look like 1e+4 instead. The number after the gap tells you how many digits follow the 1 before you reach the decimal point, so long as it's positive.

Pressing the EXP key (which I've called "xp" just to stop the button growing too big - JavaScript doesn't sppear to offer proper controls over button sizes) allows you to type in the size of the number following ^ so that you can enter really huge and tiny numbers directly.

### Pi (π)

Pi, pronounced like the word "pie" and represented by the Greek letter π, is a special number that relates the length of the circumference of a circle (the distance round its edge) to its diameter (the distance straight across it from one side to the opposite side). The value of pi is slightly more than 3, but it's impossible to write its absolute value down: it's approximately 3.14159... but the number of digits after the point goes on randomly forever. So, if you have a circle which is a metre across from one side to the other, the length round the edge will be 3.14159... metres. If you do the same thing with a square of one metre across, the length round the edge will be 4 metres. There is actually a relationship between 4 and 3.14 which you can see below.

How to calculate π:-

π   =   4/1   -   4/3   +   4/5   -   4/7   +   4/9   -   4/11   ... (and so on forever)

Here's a little program which calculates pi simply by adding and subtracting fractions in the above manner. Each time you click the "run" button it will add a fraction, store the result, then take the next fraction away from that and store that result as well (e.g. we start at 0, so the first time it runs it adds 0 plus 4/1, stores that result, then it takes 4/3 away from that and stores the new result, and then it averages those two results before displaying the answer below in red. The bottom number of each successive fraction is 2 bigger than in the fraction before, and you can watch this value grow too (in brackets) as the program runs. The real value of pi is in blue, and the values in green are popular approximations of pi using the simple fractions 22/7 and the more accurate 355/113 which replaces 22/7 as the program runs.

π     =     3.1415926535897932...        (infinity)
a     =     0.0000000000000000...        (0)
22 / 7     =     3.1428571428571430...        (43)

If you click on "run", you can then hold down the Return key to repeat the action over and over again rapidly, but it will still take a long time before it gets close to the real value of pi. To speed it up, click the following button to double the number of calculations it does for each click of "run": 1. The bottom number of the fractions has to reach about five million before the value gets as close as it will given the limitations caused by the imprecision of the fractions - for a more accurate value of pi you'd need to increase the number of digits used to represent them.

Here's some of the actual program code for those of you who want to see it:-

setup:  n=1; t=0;         (n is an initial value for the bottom number of the fractions, t is the total)

loop:   t=t+4/n;          (new total = previous total + 4/n, and n=1 the first time this code runs)
m=t;              (store the result as m)
t=t-4/(n+2);      (new total = previous total - 4/(n+2), and that means t - 4/3 the first time)
n+=4;             (change the value of n for the next time the loop runs - it'll be 5 the second time)
m=(m+t)/2         (add the results together and halve the result to get the average)
paj.innerHTML=m;  (print that number to the screen)

So, that's the loop that gets run every time you click the "run" button.

### 1/x

This key simply takes the number on the screen and divides 1 by it. It's often very useful (you can type in an eighth just by typing 8 followed by the 1/x key), but there's one particular number which is worth trying it out with, and this relates to the "golden ratio" (or golden mean), a special ratio associated with beauty. If you type the keys on a calculator as shown a couple of paragraphs below here, you'll get a number to appear on the screen which behaves in an interesting way when you press the "1/x" key repeatedly. The ratio 1 to 0.6180339 is the golden ratio, and so is the ratio 1 to 1.6180339, and all the digits after the point are the same (although the last one or two on the screen may be different due to errors).

Think about a five-pointed star, also known as a pentagram. It's a very attractive shape, and that's primarily because the golden average appears in it over and over again. The angles of the sharp corners can be worked out as follows. First imagine walking round the five straight lines that make up the star and work out how many complete rotations you will make along the way. I've given the corners the names A, B, C, D and E, so the journey I want you to imagine taking will start at A (facing towards C), then go to C, turn left and go to E, turn left and go to B, turn left and go to D, and finally turn left and return to A , though I want you to turn left again at A to face towards C so that you are facing the way you were at the start. To do this trip you will make two complete rotations by the finish, so that's 720° (2 × 360). There are five turns along the way, so that's a 144° turn at corner (5 × 144 = 720). If you were to turn to face in the opposite direction, that would be a 180° turn, so if we take 144 away from 180 we'll end up with the actual angle of each of the corners, and that's 36°.

Now that we know the angle of the corners, we can start to calculate the lengths of various lines. If I make the length of each line between two of the star's sharp corners 1 metre (so the light blue line from E to C is 1 metre long), then half of that length will be 0.5 (or 50 centimetres), and I can mark the point F in the middle of one of these lines (I've put it in the middle of the line between points A and D). I can then connect that point with a new line leading to the opposite side of the inner pentagon where it will hit the point G. The line from F to G is at 90° to the one from A to F, so I have now managed to construct a right-angled triangle with a side of length 0.5 and an angle of 36° in it's sharpest corner. That is all I need to work out the length of the dark blue line from A to G by using the function "cos": and I end up with 0.6180339, so the distance from A to G is 0.6180339 (or about 62cm). The dark blue line is 0.6180336 times the length of the light blue line, and that's the golden ratio.

Before we go any further, I should probably tell you about the usual convention for naming lines: they simply take their names from the names of the corners at either end, so the dark blue line is called AG, while the light blue line is called CE. There are two lengths which we still want to know, and the first of those is the red line CG. That's easy to work out because all we have to do is take the length of AG away from the length of AC, so that's 1 minus 0.6180339, meaning that CG is 0.381966 long. We can now work out the length of a side of the inner pentagon (e.g. the yellow line) simply by taking the length of CG away twice from the length of BD, and that leaves us with 0.2360679. Now, here's the really interesting bit: try dividing 0,6180339 by 0.381966 and see what number you get. Then try dividing 0,381966 by 0.2360679 and see what number that gives you. Yes, your eyes are not deceiving you - it's the golden ratio every time! The five pointed star is officially a very beautiful shape indeed.

(By the way, the distance between neighbouring points of the star, e.g. the line AB, is the same length as AG - you should be able to prove this for yourself by drawing a line from A to B, mark a new point H in the middle of that new line, then draw another line from H to the centre of the star, and a third line from A to the centre of the star, which we'll call S. You can then work out the length from AS in the same way the length of AG was worked out earlier, the angle at the sharp end obviously being 18° this time instead of 36°. We now have a new right-angled triange with corners at A, H and S, the right angled corner being at H, and we know the length of the side AS: this means we can work out the length of the line AH if we can find the angle of the corner at the centre of the star, and that's really easy to work out as it's clearly going to be a tenth of 360°. The last bit needs the "sin" function because it's working with the length of the opposite side and the hypotenuse (you can read up on this elsewhere on this page). Sin 36° = AH/AS. Once you've worked out the length of AH, just double it and you'll have the length of AB.)

But why is the golden ratio associated with beauty. Well, the answer is simply that it turns up over and over again in nature whenever living things develop healthily. It ties in with a special sequence of numbers known as the Fibonacci sequence: 0 1 1 2 3 5 8 13 21 34 etc. (each number being the sum of the two numbers before it, such as 5+8=13), and you can find numbers on this sequence if you count things in patterns such as the layout of seeds in a sunflower. It also represents the increase in the size of animal populations in the absense of predators and without limitations on the food supply. The ratio of any number in the sequence to either of the numbers next to it is approximately the golden ratio (apart from the first few), and it gets closer and closer to being the golden ratio as you go further through the sequence. Many of the lengths of parts of the human face and body are related by the golden ratio, and even more closely so in those individuals considered to be particularly good looking - it suggests healthy growth and therefore a particularly good person to marry and have children with. Because we find the golden ratio attractive, we like to see it in landscapes too, even though it has no relevance to how healthy the landscape is, so we use it in photography to frame things in the most attractive way.

### x!

If you press this button after typing in a whole number, it simply takes that number and multiplies it by every smaller whole number, so 6! = 6×5×4×3×2×1 = 720.

### Degrees, minutes and seconds

This key is actually more often used for hours, minutes and seconds, and it's very useful indeed. Like hours, degrees are divided into 60 minutes (though these minutes are angles and have nothing to do with time), and minutes are divided into 60 seconds, so half a degree is 30 minutes. Let's just look at it in relation to time though. If it takes you one hour twenty five minutes and four seconds to run ten miles, you can type that in as follows (make sure the button under this paragraph is visible on the screen before you start typing these): and all the hard work is done for you to convert it into hours with the minutes and seconds turning into the part after the decimal point. If you press it may be displayed as 1°25°4 or as 1°25'4'', depending on how it's been programmed and the limitations of what the screen is capable of displaying (I prefer the 1°25°4 format as it's easier to read). Pressing turns it back into a single decimal number. If you then divide by 10 and press you will see your average time for each mile as 0°8°30.4 (just over eight and a half minutes), which is far easier to understand than the 0.1417777 which you would see otherwise.

Here are some useful formulae for runners, cyclists and swimmers. The first two are for working out what time you should achieve if you race a particular distance at a set speed, while the second two are for working out your speed from the distance and your time:-

miles (enter distance) mph (enter speed) (displays the time).

km (enter distance) kmph (enter speed) (displays the time).

mls hours minutes seconds (displays speed in mph).

km hours minutes seconds (displays speed in kmph).

### Inverse or 2nd Function Key

On a calculator is is normal to have more than one function on a key in order to reduce the number of buttons and size of the machine, but in many cases the second function does the same thing as the main function of the key but in the opposite direction (e.g. the inverse of sin is asin, so you can get the asin function by pressing "inv/2nd" followed by "sin"). In the calculator on this page you can manage without using the "inv/2ndf" key as there are always buttons here for both directions, though if you're using the keyboard for speed you need to use "i" before "sin", "cos" and "tan" for "asin", "acos" and "atan".

### Sine (usually shortened to sin)

Sin, pronounced like the word "sign", is useful for working out angles and lengths. The most common use of it is with right-angled triangles like the one in the diagram below, and the rule is: Sin z° = Opposite ÷ Hypotenuse (the line from X to Y is opposite the angle z, so it's called "O" for "opposite", while the line from X to Z is is called "H" for "hypotenuse", the name of the longest side of any right-angled triangle). If the angle z is 30° and the length of the opposite side "O" is 20cm, then we can work out the length of H (the hypotenuse) simply by dividing 20 by the sine of 30°: . Feel free to try out alternative values for the angle z.

But what happens if you don't know the length of O to start with, but do know the length of H? Well, you can work out the size of O by multiplying the sine of z by H. If H=40 and z=30°, this is what you should type in: .

And what if you don't know how big the angle z is, but do know the lengths of O and H? Well, all you need to do is divide the length of O by H and then press the "asin" key (or "inv" followed by "sin") which does the reverse action of the sin key. If you know that O=20 and H=40, you can get the size of angle z as follows: .

### How to remember how to use sin and asin

The important thing to memorise is this: sin z° = O/H, or just remember the word SOH which is part of a mantra which you might like to try chanting: SOH CAH TOA, SOH CAH TOA, SOH CAH TOA, SOH CAH TOA... (the CAH and TOA parts refer to the cos and tan functions which work the same way but using different sides of the triangle). The word SOH means that the function Sine uses Opposite/Hypotenuse. Once you have that idea in your head you can turn it into sin z° = O/H and then replace the letters with the known values. If you know the lengths of O and H, you can simply divide O by H (remember that O is a letter, not the number 0) and then use asin to calculate z. If you already know the value z but only know one of the lengths, you can type the angle value in and press "sin" to simplify that side of the equation, and then you have to use your equation manipulation skills to rearrange the equation, the idea being to get the unknown value on its own at one side of the "=" sign while the known values are all on the other side. We initially have sin z° = O ÷ H. The only option available for the first move is to transfer the ÷ H to the other side of the "=" and change the "÷" into "×", so we now have sin z° × H = O. If the unknown length is O, it can now be worked out easily, but if the unknown length is H, we have to make another move to isolate H. We can rephrase the equation as H × sin z° = O, and then we can move the × sin z° part to the other side of the equation and change the "×" back into "÷": H = O ÷ sin z°.

### Cosine (usually shortened to cos)

Cos is useful for working out angles and lengths. The most common use of it is with right-angled triangles like the one in the diagram below, and the rule is: Cos z° = Adjacent ÷ Hypotenuse (the line from X to Z is adjacent to the angle z, the word "adjacent" meaning "connected to", so this line is called "A" for "adjacent", while the line from X to Z is is called "H" for "hypotenuse". If the angle z is 30° and the length of the adjacent side "A" is 40cm, then we can work out the length of H (the hypotenuse) simply by dividing 40 by the cosine of 30°: . Feel free to try out alternative values for the angle Z.

But what happens if you don't know the length of A to start with, but do know the length of H? Well, you can work out the size of A by multiplying the cosine of z by H. If H=46 and z=30°, this is what you should type in: .

And what if you don't know how big the angle z is, but do know the lengths of A and H? Well, all you need to do is divide the length of A by H and then press the "acos" key (or "inv" followed by "cos") which does the reverse action of the cos key. If you know that A=40 and H=46, you can get the size of angle z as follows: .

### How to remember how to use cos and acos

The important thing to memorise is this: cos z° = A/H, or just remember the word CAH which is part of a mantra which you can chant: SOH CAH TOA, SOH CAH TOA, SOH CAH TOA, SOH CAH TOA... (the SOH and TOA parts refer to the sin and tan functions which work the same way but using different sides of the triangle). The word CAH means that Cosine uses Adjacent/Hypotenuse. Once you have that idea in your head you can turn it into cos z° = A/H and then replace the letters with the known values. If you know the lengths of A and H, you can simply divide A by H and then use acos to calculate z. If you already know the value z but only know one of the lengths, you can type the angle value in and press "cos" to simplify that side of the equation, and then you have to use your equation manipulation skills to rearrange it, the idea being to get the unknown value on its own at one side of the "=" sign while the known values are all on the other side. We have cos z° = A ÷ H. The only option available for the first move is to transfer the ÷ H to the other side of the "=" and change the "÷" into "×", so we now have cos z° × H = A. If the unknown length is A, it can now be worked out easily, but if the unknown length is H, we have to make another move to isolate H. We can rephrase the equation as H × cos z° = A, and then we can move the × cos z° part to the other side of the equation and change the "×" back into "÷": H = A ÷ cos z°.

### Tangent (usually shortened to tan)

Tan is useful for working out angles and lengths. The most common use of it is with right-angled triangles like the one in the diagram below, and the rule is: Tan z° = Opposite ÷ Adjacent (the line from X to Y is opposite the angle Z, so it's called "O" for "opposite", while the line from Y to Z is is called "A" for "adjacent", this line being adjacent to (or connected to) the angle at Z. If the angle z is 30° and the length of the opposite side "O" is 20cm, then we can work out the length of A (the adjacent side) simply by dividing 20 by the tangent of 30°: . Feel free to try out alternative values for the angle z.

But what happens if you don't know the length of O to start with, but do know the length of A? Well, you can work out the size of O by multiplying tangent of z° by A. If A=35 and z=30°, this is what you should type in: .

And what if you don't know how big the angle z is, but do know the lengths of O and A? Well, all you need to do is divide the length of O by A and then press the "atan" key (or "inv" followed by "tan") which does the reverse action of the tan key. If you know that O=20 and A=35, you can get the size of angle z as follows: .

### How to remember how to use tan and atan

The important thing to memorise is this: tan z° = O/A, or just remember the word TOA which is part of a mantra which you can chant: SOH CAH TOA, SOH CAH TOA, SOH CAH TOA, SOH CAH TOA... (the SOH and CAH parts refer to the sin and cos functions which work the same way but using different sides of the triangle). The word TOA means that Tangent uses Opposite/Adjacent. Once you have that idea in your head you can turn it into tan z° = O/A and then replace the letters with the known values. If you know the lengths of O and A, you can simply divide O by A and then use atan to calculate z. If you already know the value z but only know one of the lengths, you can type the angle value in and press "tan" to simplify that side of the equation, and then you have to use your equation manipulation skills to rearrange it, the idea being to get the unknown value on its own at one side of the "=" sign while the known values are all on the other side. We have tan z° = O ÷ A. The only option available for the first move is to transfer the ÷ A to the other side of the "=" and change the "÷" into "×", so we now have tan z° × A = O. If the unknown length is O, it can now be worked out easily, but if the unknown length is A, we have to make another move to isolate A. We can rephrase the equation as A × tan z° = O, and then we can move the × tan z° part to the other side of the equation and change the "×" back into "÷": A = O ÷ tan z°.

Calculators have three different modes for dealing with angles. The normal one is to use degrees like on a compass, so there are 360 degrees in a circle and a right angle is 90°. The French army use gradians instead, so they have 400 of them in a circle and 100 in a right angle. The other system, radians, seems odd at first sight because there are 6.283185307179586 (2π) of them in a circle: one radian = 57.29577951308232°. Why would anyone use such a strange unit for measuring angles? Well, it's simply that it's often more convenient than degrees in mathematics, and also more convenient for computers - they always work with radians and then convert into degrees. The following is a way to work out the sine of an angle, and the key point is that it only works if you express the angle x in radians:-

sin(x)   =   x^1/1!   -   x^3/3!   +   x^5/5!   -   x^7/7!   +   x^9/9!   -   x^11/11!   ... (etc.)

Don't be scared of the strange symbols: there's nothing difficult about them. The symbol ^ means that you have to multiply the number before the ^ by itself as many times as stated by the number following the ^, so   4^3 = 4×4×4;   4^5 = 4×4×4×4×4.   The symbol ! in mathematics is also just a shorthand for something very simple:   1! = 1;   2! = 1×2 = 2;   3! = 1×2×3 = 6;   4! = 1×2×3×4 = 24.

Armed with that little bit of knowledge, I have written a little program which calculates the sine of an angle simply by adding and subtracting fractions in the above manner. Each time you click the "run" button it will add a fraction, store the result, then take the next fraction away from that and store that result as well (e.g. we start at 0, so the first time it runs it adds 0 plus x^1/1!, stores that result, then it takes x^3/3! away from that and stores the new result, and then it averages those two results before displaying the answer below in red. The value that grows by 2 in each successive fraction is shown in brackets as the program runs. The real value of sin(x) will be shown in blue so that you can watch the value in red approach it as it gets more accurate.

x=30°
sin(x)     =     0.5
a     =     ...
(0)

It only takes a few clicks on "run" before an accurate value is reached.

Here's some of the actual program code for those of you who want to see it:-

setup:  t=0; n0=1; n1=1; ang=0.5235987755982988;   (t = total, n1 = 1/3/5/7 sequence value, n0 = n1! total,
ang = 30° converted into radians)

loop:   t=t+Math.pow(ang,n1)/n0;        (new total = previous total + angle^n1/n0)
n1+=1; n0*=n1; n1+=1; n0*=n1;   (prepare n1 and n0 for next fraction)
m=t;                            (store the result as m)
ts=ts-Math.pow(ang,n1)/n0;      (new total = previous total - angle^n1/n0)
n1+=1; n0*=n1; n1+=1; n0*=n1;   (prepare n1 and n0 for next fraction)
m=(m+t)/2                       (add the results together; halve the result to get the average)
sn.innerHTML=m;                 (print that number to the screen)

So, that's the loop that gets run every time you click the "run" button. [Notes: n+=1 means n=n+1; n*=1 means n=n×1.]

### Square Root and Squared

If after typing in a number you press the "root" button, the calculator works out which number times itself equals the number you started with, so if you type in 16 followed by "root", it will give you the answer 4 (because 4×4=16). If you know that a square has an area of 16 square centimetres, the square root of 16 will be the length of the side of that square in centimetres (4cm). The normal symbol for a square root is:-

` / _ 9 = 3

I would use that symbol more often, but there's no easy way to type it in or to display it on the screen - I had to draw the above by combining ` / and _. The number within the symbol, 9 in this case, is the one to be processed by the square root function, and the square root of 9 is 3 because 9=3×3.

The key with x^2 on it is for doing the opposite, the "x" representing the number you type in before pressing this button, so typing in 4 followed by x^2 will give the answer 16. On most calculators you will see x2 instead of x^2, but it means exactly the same thing (the ^ symbol is frequently used on computers instead as it's easier to type in - the upward pointing arrowhead simply indicates that the number following it would be written higher up if the ^ was left out). The normal way to write things out on paper is as follows:-

3² = 9

Now it's time to look at how these functions can be used. Thousands of years ago, a Greek by the name of Pythagoras found a way to prove that the longest side of a right-angled triangle (that's a triangle in which one of the corners is 90°) is always related to the lengths of the shorter two sides in a particular way. First let's look at a diagram so that you can understand what this is about:-

The short sides have been given the names "a" and "b", while the long sloping side is called "c". These names can also be used to represent the lengths of those sides, so we can say that a=3 and b=4. The question now is, how long is c? Well, the rule is:-

a² + b² = c²

We know the values of a and b, so we can now turn the above into 32 + 42 = c2, and then we can do the squaring to turn it into 9 + 16 = c2. We can now simplify it further by adding the 9 to the 16, so that gives us 25 = c2. To work out what c is, we need the square root of 25, and that is of course 5 (because 5×5=25). This rule works with any shape of right-angled triangle, and Pythagoras managed to prove it with this stunning visual proof:-

In each of these two diagrams you can see four triangles of the same shape and size contained within a much larger square. In the first diagram there is a smaller white square in the middle, while in the second diagram there are two white squares. The white area (without the criss-cross lines) in each diagram must be the same size because the large square containing the triangles in each diagram are the same size and all the triangles are the same size as each other. Now think about what squaring a number can do: you can use it to find the area of a square from the length of its side. In the first diagram we have a square which has the same length of side as the longest side of the original triangle (the one called c). In the second diagram we have two white squares, one of which has the same length of side as the shortest side of the original triangle (the one called a), while the other square has the same length of size as the remaining side of the triangle (the one called b). We can get work out the areas of the two white squares in the second diagram by squaring the lengths of a and b, then we can add them together to work out the area of the white square in the first diagram, and finally we can take the square root of that to work out the length of its side, and that will be the same as the length of the longest side of the triangle (the side called c). If you change the shape of the original triangle, this will still work in exactly the same way, just so long as it is a right-angled triangle, so it is clearly always true with a right-angled triangle that by adding the squares of the shorter two sides, you get the square of the longest side.

### Roots:   x^1/y   (also known as:   x1/y )

With this key you have to type in a number first, then press , and then type in a second number to determine what kind of root you want (while pressing the equals key produces the answer). If this second number is a 2, you'll get a square root, whereas a 3 will give a cube root.

If you know that a square has an area of 16 square centimetres, the square root of 16 will give you the length of the side of that square in centimetres (4cm). The question is 16 = what × itself, and the answer is 16=4×4.

If you know that a cube has a volume of 64 cubic centimetres, the cube root of 64 will give you the length of any edge of that cube in centimetres (4cm). The question this time is 64 = what × itself × itself, and the answer is 64=4×4×4.

### Powers:   x^y   (also known as:   xy )

With this key you have to type in a number first, then press , and then type in another button to determine what power of a number you want (while pressing the equals key produces the answer). If this second number is a 2, the first number will be squared, so if you know that the edge of a square is 3cm and you want to know its area, typing in will produce the answer 9 square centimetres (3×3=9). (By the way,none of the buttons in this paragraph are wired in, so they don't work.) If the length of an edge on a cube is 3cm, you can work out the volume of a cube by pressing and the answer 27 cubic centimetres will be produced (3×3×3=27).

It's easy to understand how powers work with whole numbers as 4^2 = 4×4 and 4^3 = 4×4×4, but you can also have 4^2.5 even though it can't be translated into a simpler form. The key to understanding this is to use the ln key (natural log), because the inverse log of (log 4 × 2.5) gives you the same answer as 4^2.5 (ignoring tiny errors). The log key works for this too, but the ln key is more accurate as it misses out a couple of extra translation stages.

### Log

The first thing you should know about the log key is that it can do ordinary things in a strange way. Take the sum 5 × 6 = 30 for example: type in , then and you'll get the answer 30 without doing any multiplying. You can multiply numbers together by adding their logs together instead. People who needed to do lots of boring mutliplying work before the days of calculators used to use log tables so that they could do everything by adding and subtracting instead of multiplying and dividing as it was a little faster (think about doing long division with multi-digit decimals) and they were less likely to make mistakes.

To give you an idea of how logs work, here are some numbers (to the right of the dashes) and their logs (to the left of the dashes):-

6 − 1000000
5 − 100000
4 − 10000
3 − 1000
2 − 100
1 − 10
0 − 0
-1 − 0.1
-2 − 0.01
-3 − 0.001
-4 − 0.0001
-5 − 0.00001
-6 − 0.000001

You can use logs to solve interesting problems such as this one: 2^what = 8. If we want to know the answer to 2^3 = what, we simply use the x^y key; if we want to know the answer to what^3 = 8, we simply use the x^1/y key, but neither of those keys can be used to calculate the answer to 2^what = 8. Fortunately, the log key comes to the rescue, because all we have to do is divide the log of 8 by the log of 2: and the answer 3 will appear. (The buttons in this paragraph aren't wired in, so pressing them does nothing.)

The above works just as well with the "ln" key (natural log, using a base 2.7182818284590455 rather than 10 - don't worry about what that means for now), and indeed it is actually better to use the "ln" key because the log key does all it's work using the "ln" function anyway - it just adds on an extra translation phase which you don't always need. Where you might still want to use the log key rather than the ln key is if you're converting absolute values to a logarithmic scale using 10 as its base such that 4 represents something ten times bigger than 3 and where 5 represents something ten times bigger than 4. Logarithmic scales are used where the range of values is enormous, the absolute values going from miniscule to astronmomical such that the number of digits in the numbers makes them hard to work with, so for earthquake strengths and sound volume we use logarithmic scales. In the case of sounds, we use decibels, though a decibel is actually a tenth of a bel, and it's the bel that fits with the log key. A sound of 100 decibels (or 10 bels) is ten times louder than a sound of 90 decibels (9 bels). A sound of 110 decibels is a hundred times louder than 90 decibels. 120 decibels is 1000 times louder than 90 decibels. 130 decibels is 10,000 times louder than 90 decibels. 140 decibels is 100,000 times louder than 90 decibels. Now think about this: exposure to 90 decibels for an hour is already more than loud enough to damage your hearing.

The Richter scale is used as a measure of earthquake magnitude (actually another scale has replaced it, but the values are so similar that most people just call it the Richter scale anyway). It also works best with the log key (rather than the ln key) as the amplitude of the shaking waves is ten times greater for a magnitude 7 quake than a magnitude 6. The destructive power, however is further multiplied such that the real strength of a magnitude 7 quake is 31.6 times greter than a magnitude 6, so that's a further complication. This explains how you can sleep through a magnitude 3 quake and never know it happened, whereas a magnitude 6 quake may knock your house down - the strength of the shaking isn't three times as much, nor even 1000 times as much, but nearly 95,000 times as great. The truth of it is, these logarithmic scales actually make it much harder for ordinary people to understand the relative strengths of things as they are really designed for use by experts.

### Natural Log: ln

The "ln" you see on the key is not the word "in", but is actually "LN" written in small letters. Natural logs are proper logs using e as their base (e = 2.7182818284590455) rather than 10 (see above to learn about what logs are about) - calculators and computers do all their log calculations in relation to the base e and then convert the results to standard logs if you're using the log key instead of the ln one. The actual conversion is done simply by dividing the natural log of a number by the natural log of 10.