If you are anything like me maths when I got to secondary/high school was a subject that was defined by two main things:

Maths is …

1. Boring and hard – it didn’t apply to the world I was living in was going to be living in. I could see that I may have to know something to do with right sided triangles aka trigonometry but none of that sockatoea thing and certainly will never use a simultaneous equation or work out a quadratic equation especially where the result was a negative number!
2. Compulsory – you had to take it. Leaving school without maths is total failure where you will never get a job or lead a fulfilled life or you would have to join the army.

And I was actually ok at maths hence the reference to simultaneous equations and the mystery of quadratic equations. I understood that if you did want to go on to higher things (which I did) then you need to at least appreciate how maths is an important tool. However, it was always a challenge, I never really got it like others in the class that seemed to have a Will Hunting level of understanding straight away. Maths is something that is fundamental to not only understanding the world but also in the new technologies of the future so I’m going to have a go at working through it at a pace that I understand and hopefully you will find help as well.

Understanding Numbers

Natural Numbers: counting the world

Maths is performed by all sorts of people in their lives many times coincidentally so the point where if you ask someone if they are good at maths they say “No” or “I get by with the basics”. And this is true humans and a few other animals naturally use a form of mathematics to get by. For example if you said to a person how many apples are there in a basket? They could count them individually or they could estimate how many there are with varying degrees of precision starting from “more than one” to “between 6-10” depending on what they could see. We can also do maths without seeing. We can tell if one thing is heavier, hotter, more or less attractive than something else. At the fundamental level we use mathematical things naturally as we are social animals so it’s good to know how to count (addition) how much food there is and how much (division) you should get.

Moving away from the cave and the stone axes into the modern era we start to use maths in two ways: 1) planning and 2) scale. From planning as we look to slightly adapt the four Fs of human kind evolution human kind 1) field, 2) farm, 3) food, 4) fight. These four Fs forced more complex maths onto people. The simple farmer needed numbers to planning and planting of seeds for the next season, manage fuel in the form of wood and how to budget resources across cold winters or very dry summers. Kings and Queens through their courts used maths to organise trade and at times coordinate both defence and attack of war. As humans connected more through faster movement of people and information time became more important not so people knew what time it was but ‘where’ time it was through navigation.

Today many of us use maths for what seems simple things in the same way humans have done for thousands of years – controlling our lives. To do this we start with the very basics of maths consisting of three main parts:

Basics of Maths contains

1. Numbers
2. Comparisons or evaluators (things that compare numbers)
3. Operators (things that change numbers)

When we talk about numbers 99.99% of use is through Natural numbers or numbers that are used by nature. These numbers are positive whole numbers, or more specifically whole integers (where integer is Latin for whole or untouched in (non) – tangere (touch)) from 1, 2, 3, … infinity. Using natural numbers we can do 99.99% of things we need to do as we are working with real things – you can count real apples even in the future if you were trying to work out how many you could store over winter. There would be no need to work out fractions of an apple or even a negative apple! It is to this point the very practical use of numbers. Natural numbers don’t have a zero as it has literally no value – if you had four apples and four apples were eaten you wouldn’t have zero apples – you’d have nothing but I’ll come to the important and problem of zero in a while.

The second part of basic maths is comparisons or evaluators. We do this very naturally: this is more than (>) that, this is less than (<) that, this is equal to (=) that. This is a critical piece to realise: equals is normally framed as ‘the answer to’ as in the answer to 1 + 1 is 2, like ‘ta da!’ as we are taught to find the answer after the equals sign: x + y = what?

A lot of the time in life we don’t need to have equals but more or less. For example how many apples are needed to make a medium apple tart? Around 10? Being good at estimating is a part of maths that we use but don’t write down as the maths we are taught is precision – things equal other things exactly.

The final part of basic maths are the operators. There are many operators but the basics ones you will already know

1. Addition – adding up /together, symbol +, result = sum e.g. the sum of 2+1 is 3
2. Subtraction – subtracting /taking away, symbol -, result = difference e.g. the difference of 2-1 is 1
3. Multiplication – multiplying, symbol x, result = product e.g. the product of 2 x 3 is 6
4. Divide – divide by, symbol ÷ , result quotient e.g. the quoitent of 6 ÷ 3 is 2

These four maths operations will get you a long way through life especially if you live by yourself. Further calculations are needed when we start to scale and share.

Natural numbers: 1, 2, 3, 4, … forever

Balancing Numbers: negative numbers

Natural numbers come up short when we need to borrow something of someone, a debt that needs to be record and then balanced out. For the most part it’s easy to do with natural numbers – I have 4 apples, you want 2 apples, I lend you 2 apples, you owe me 2 apples. But if we wish to track this we need to mark the borrowed apples from the real apples to show who and when they belong. In accounting we do by having one line or book that has all the money that is owed and one that we owe. These have to balance to make sure that the business accounts are net zero as the monetary value of the business is actual owed to the business owners.

For the apple example in simple accounting it would start with me owning 4 apples and your account (or ledger as that is what it is physically recorded in) having 0. When I lend you 2 apples my account now has a record that I still have 4 but 2 are lent out so still 4 and yours would have 1 record for the 2 apples you have and a negated record saying they belong to me so need to be paid back

My Apples

4: 2 with me, 2 with you

Your apples

0: 2 with you, -2 owed to me

This is where the idea of the need for negative numbers is needed. In reality we are still using natural numbers but we class them as the opposite so they are the opposite – in effect negative apples! Again this is simple maths that you may (should) do to balance the money you make (income) with the money you spend (expense). At the end of the month these two states need to balance – cancel out hence the phrase balance the books.

We now have natural numbers and negative numbers. We also introduced a symbol for nothing – zero which is what we get when we add the positive number (specifically positive integer) to the same size but negative number: 5 – 5 = 0 and -5 + 5 = 0.

Negative Numbers: -1, -2, -3, -4, … forever

Rational Numbers – being divisive

The next type of number is one you have also done in school – fractions where a number is expressed as one integer over or on top (numerator) of another integer (denominator) 1/3 or 5/7 (numerator /denominator). There are three types of fraction:

1. Proper fraction: numerator smaller than denominator e.g. 1/2
2. Improper fraction: numerator bigger than denominator e.g. 5/4
3. Mixed fraction: a whole number and a proper fraction e.g. 1 1/4

Only a proper fraction is, well, proper as it represents a the most simplified state of the number. In effect it represents part of a whole number – a fraction of it or a ratio of a whole number 1:2 means 1 part of something to 2 parts of something or half of something to 1 of something. Fractions are called rational numbers not because they and logical and calm but because they represent a ratio of a number (fun fact: rational numbers didn’t come from ratio but ratio came rational which in turn came from irrational numbers which I’ll cover in a bit).

But what’s the point of saying 1/3 when I can use 0.33333 for my calculations? Answer is absolute precious and balance.

0.3333333333333 ≠ 1/3

It is very close and not the vast majority of calculations it will do but it’s not equal (we will talk about this if you get the section on limits!).

So far so good. Numbers, or integers, can be positive (1, 2, 3, 4, …) or negative (-1, -2, -3, -4, …) with zero (0) included. We can also use fractions ( 1/3, 1/ 1000) giving us 99.99% of numbers (9999/100) that we use.

But there are more, exotic numbers that most do not recognise as a number as it is not expressed as integers like above. Instead the number is expressed as either symbol or a symbol and an integer.

Rational Numbers: 1/1, 1/2, 1/3, 1/4, … and the negative fractions

The Irrational Numbers – numbers as symbols

In the world of the obvious and logical rational numbers are ones that can be expressed a ratio (1:3 or 1/3) and those that can not must be irrational. Simple. However a ratio and a fraction are not the same thing – one expressed proportions e.g. the door is 2 metres by 1 metre or 5 parts flour to 1 part sugar; fraction expression how much of a whole e.g. I ate half the cake, or I’ve driven two-thirds of the journey. The reason rational numbers is not because they are logical and proportional but they are not weird, odd, and un-godly. I say un-godly as the famous Greek mathematician Pythagoras believed that every number could be expressed as a fraction as the universe has order through repeating patterns. When it was proved that, unfortunately, that the square root of 2 has no fraction it was deemed irrational – ta da – irrational numbers (it is said that Hippasus of Metapontum, a follower of Pythagoras was drowned after he proved the square root of 2 was irrational as it was seen as heresy (probably should have kept it to himself).

You may think I’m never going to use irrational numbers and you are probably right. If you do come across one it is likely to be pi or π the lower case form of 16th letter of the Greek alphabet. Pi is used in calculating both circumferences of circles through multiplying the diameter (d) of the circle (or multiplying the radius by 2 as the radius is half the diameter) by pi (πd), or calculating the area by multiplying radius by itself (squaring the radius) then multiplying by pi (πr^2). Even in these circumstances you won’t actually be using pi as the calculator you are using can only hold it to a limited number of say 9 digits.

Irrational Numbers: π (pi), √2 (square root of 2)

Real Numbers – all of the above

All the above number sets are used in reality. They are called Real Numbers for this reason. For 99.9999999% of the populations this is all they need to know. However, to complete the story on numbers we need to use our imaginations and make things a tiny bit complicated.

Imaginary and Complex Numbers

I won’t spend too much time on imaginary and complex numbers because a) you don’t need to use them and b) I don’t really understand them that much. I will cover them because they are amazing and it completes the story on numbers.

Imaginary numbers are a result of having to find the square root of a negative number. Squaring a number is simple: multiply a number by itself, or raise it’s power by 1. It’s called squaring as you can draw it as a square with 2 dimensions the same (don’t confuse having a square number with a number squared).

sidebar …

Square metres vs Metres Square – confusion with squaring

A classic example of maths taught at school hitting the reality of life is the confusion between square metres and metres squared. This is in part due to poor English rather than maths but it’s worth explaining the difference.

Imagine you are looking to cover a floor with carpet and you want to know how much carpet to buy. The floor is 3 metres by 3 metres. Using your school maths you know it is 3m squared (3 x 3 = 32) in the same way that any number multiplied by itself is the number squared. OK – all good. Metres squared.

You go to the carpet shop and find a nice carpet for £10 per square metre. Do you 3 square metres? Nope. You need 9 as the area is made up of 9 x 1m2 squares. Squares of 1m x 1m.

Remember that in the real world you are unlikely to ever use a unit squared. Find the total area and buy that amount in square metres.

OK. Back to the imaginary numbers. We need to appreciate two things to have a handle on imaginary numbers.

1. No such thing as a negative square

There is an odd thing in mathematics when it comes to multiplication and that is how it handles negative numbers. The rules are the following:

• positive x positive = positive -> 2 x 2 = 4
• positive x negative = negative -> 2 x -2 = -4 (2 lots of -2)
• negative x positive = negative -> as previous
• negative x negative = positive -> -2 x -2 = 4

The negative x negative resulting in a positive seems like an odd result but it’s provable and for now just accept it. What this means is there is no such thing as a negative square as both 2 and -2 squared result in 4 (if you though -2 x 2 = -2², nice try but -2 and 2 are not the same number).

2. Any square number has two roots

The opposite of squaring a number is rooting or finding the root of the square. I can’t picture a scenario in everyday life where you would need to find the square root of anything so it’s likely limited to those doing maths in your profession (which you may be).

To work out the root of any number we need to find the single number when multiplied together gives that number. For lots of numbers we can work this out even if it takes a little time or use a calculator. For example the number 9. We know that 3 x 3 is 3² or 9. This means that 9 divided by 3 is 3 so 3 is the square root. So far so good.

Where it gets a little trickier. Remember that a negative multiplied by a negative is a positive then the square root of 9 is 3 AND -3 as -3 x -3 = 9. All numbers have two square roots to cover the positive and negative e.g. the square root of 9 is 3 and -3.

But the number -9 does exist so what is the square root? And does it matter? Here is where imaginary numbers come into existence to hold some complicated maths together. To cut a long story short the square root of -9 is 3i (for a fuller and brilliant explanation watch Veritasium video on imaginary numbers). Imaginary numbers came from the idea that we have to imagine that things exist but in a plain that normal numbers don’t work. They help to glue together maths where conventional numbers don’t work in the same way that we use negative numbers where we have to record something past a positive point (like a bank account or temperature or coordinates).

There is a little more to this and that is every imaginary number has a core factor of the square root of -1 or √-1 in the same way that every Real number has a factor of 1. Therefore we express an imaginary number as just i to represent √-1.

Imaginary numbers: 1i, 2i, 3i, 4i, ... √-1

Complex Numbers

The last number type is complex numbers. Complex numbers are combination of a Real Number and an Imaginary number which has an additional factor or component or magnitude. At this stage it’s crazy numbers which I’ll let you find out more about as I’m not qualified to really comment.