A Guide to A-Level Maths Hello, young adventurer! Welcome to a world of abstractness. But don't worry, here's an illustrated student guide book... written by a student.
However, as you could probably tell, this book is no where near finishing. You might be able to help the author speed things up by showing some encouragement. You never know, clicking that star at the top-right corner might help.
If the following button is green, this book is up and running without major problems! book passing
2
Chapter 1 Surds And Indices In the beginning, you learnt how to add ``one'' to a number. Three becomes four, four becomes five, and so on. Then you've learnt addition, where you repeat adding 1s to a number. One day, you learnt that you can repeatedly add a number, which you now know as multiplying: \[a \times n = \underbrace{a+a+a+\dotsb+a}_{\text{\(n\) times}}\] Can you repeat multiplication? Sure! This is what this chapter's about.
Types of Numbers Some people have their lucky numbers. Some people have numbers which they hate. Some numbers are even colourful to some people. We, people, put numbers into categories, just like anny other thing. However, there are some important categories of numbers you should know: Integers, Rational Numbers, Irrational Numbers, and Real Numbers. We group these numbers using sets.
Integers Integers are whole numbers, both positive and negative including zero. Integers do not have a fractional part (e.g. doesn't have anything after the decimal point, another way of saying that they are whole numbers). Example: the numbers \(1\), \(5\), \(0\), and \(-234\), are all integers, but the numbers \ (2.3\), \(\tfrac{4}{5}\), \(-\tfrac{3}{11}\) are not integers.
3
Set of Integers Integers can be grouped together into a set. We call this as the set of integers, labelled as \(\mathbb{Z}\) for the German word Zahlen meaning ``numbers''. \[\mathbb{Z} = \{\dotsc, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dotsc\}.\] If a number, \(x\), is an integer, then \(x \in \mathbb{Z}\). For example, \(-4 \in \mathbb{Z}\) because negative four is an integer. If \(x\) is not an integer, then \(x \notin \mathbb{Z}\). For example, \(\tfrac{1} {3}\notin\mathbb{Z}\) because a third is not a whole number. Also, \ (\text{donkey}\notin\mathbb{Z}\) because a donkey is not even a number!
Rational Numbers A rational number is a number that can be written as a fraction of two integers. All integers are also rational numbers, because integer \(x=\frac{x}{1}\), which is a fraction of two integers. Also: a number is rational if: the decimal part terminates (ends somewhere like ``3.25'') or the decimal part repeats indefinitely (e.g. ``2.135135135...'', because they can be written as a fraction of two integers.
4
Examples Rational numbers: \(3\), \(\frac{2}{5}\), \(-\frac{3}{8}\), \(5.64\), \ (7.123123123123\dotsc\). Not rational numbers: \(\sqrt{2}\approx 1.41421356\dotsc\), \(\pi \approx 3.14159265\dotsc\), \(e \approx 2.718281828459\dotsc\). Their decimal part does not repeat. They are called irrational numbers. The \\(\approx\\)'' sign (above) meansis approx. equal to'', as opposed to the equal sign.
5
Set of Rational Numbers The set of rational numbers contains every number that can be expressed as a fraction of two integers. We label it with \(\mathbb{Q}\) for quotient, another way of saying ``fraction''. The set of integers is a subset of the set of rational numbers, because \(\mathbb{Q}\) contains all integers as well as other numbers. The set of positive rational numbers can be represented with the symbol \ (\mathbb{Q}^+\). The set of negative rational numbers can be represented with the symbol \ (\mathbb{Q}^-\).
Irrational Numbers Real Numbers Watch this space guys!
Raising Powers If you multiply a number by four, you add the original number to \(0\) four times. When you raise a number to the \(n\)th power, you multiply \(1\) by the number four times. We write \(b^n\), and say ``\(b\) to the power \(m\)''. This way of writing is known as index notation. \[b^n = \underbrace{b \times b \times b \times \dotsb \times b}_n.\]
6
Squares Squaring a number means multiplying itself by itself, so the square of \(x\) is \(x^2\) and we say ``\(x\) squared''. It's useful to see that any number, when squared, will always be positive or zero.
7
Cubes Similarly, cubing raises a number to the power of three.
8
Naming This Operation Just as multiplying is called multiplication, raising a number to a power is called exponentiation.
Index Notation Index notation is a way of writting powers.
Note: the plural of index is indices
Using index notation, \[ \color{#009ce4}{ \underbrace{ \color{#009828}{a \times a \times a \times \dotsb \times a} }_{\text{\(m\) times}} } = \color{#009828} {a}^{\color{#009ce4}m} \,, \] where \(\color{#009828}a\) is the base, and \ (\color{#009ce4}m\) is the index (also known as the exponent and the power). We say \\(a\\) to the power \\(m\\)'', or even just\(a\) to the \ (m\)'' since it's easier to say out loud.
Multiplying Expressions That Contains Exponents If you multiply two numbers with the same base together, you'll find that \[\color{#a31500}{a^m} \times \color{#009828}{a^n} = a^{\color{#a31500} {m}+\color{#009828}{n}}\,.\]
9
This is because \begin{align} \color{#a31500}{ a^m }&= \underbrace{ \color{#a31500}{ a \times \dotsb \times a } }_{ \color{#a31500}{ \text{\(m\) times} } } , && \text{and}\\ \color{#009828}{ a^n }&= \underbrace{ \color{#009828}{ a \times \dotsb \times a } }_{ \color{#009828}{ \text{\(n\) times} } } \end{align} so if you multiply them together, you will get \begin{align} \color{#a31500}{a^m} \times \color{#009828}{a^n} &= \overbrace{ \color{#a31500}{ a \times \dotsb \times a } }^{ \color{#a31500}{ \text{\(m\) times} } } \times \overbrace{ \color{#009828}{ a \times \dotsb \times a } }^{ \color{#009828}{ \text{\(n\) times} } } \\ &= \underbrace{ \color{#a31500}{ a \times \dotsb \times a } \times \color{#009828}{ a \times \dotsb \times a } }_{ \text{\(m+n\) times} } &=a^{m+n}\,. \end{align}
10
Just Give Me an Example Multiply \(2^6\) and \(2^4\) together. You'll get something like \[2^6 \times 2^4 = \underbrace{\overbrace{2\times2\times\dotsb\times 2}^{\text{\(6\) times}} \times \overbrace{2\times\dotsb\times 2}^{\text{\(4\) times}}}_{\text{\(10\) times}} = 2^{10}\,.\]
Dividing Expressions That Contains Exponents You saw that multiplying adds the number of things being multiplied, increasing the exponent. When you divide, the opposite happens: \[a^m \div a^n = a^{m-n} \,.\] This is because \(a^n\) cancels \(n\) lots of \(a\) from \(a^m\), so for example, \[ \frac{2^6}{2^4} = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2} = 2 \times 2 \times \frac{2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2} = 2^2\,.\] Still following along? (If not, leave a note).
Raising a Power of Another Power. The title means that if you raise a number to a certain power, and then use raise the resulting number to another power, what do you get? In symbols, we get \[\left(a^m\right)^n = a^{mn}\,.\]
11
Explanation The \(a^m\) expression inside means this: \[\underbrace{a \times a \times \dotsb \times a}_{\text{\(a\) appearing \(m\) times}}\,.\] The \(({\phantom{a}})^n\) part means to repeat ``\(a^m\)'' \(n\) times: \[ \text{repeat \ (n\) times} \left\{ \vphantom{ \begin{align} &a\\ &a\\ &\vdots\\ &a \end{align} } \right. \begin{align} &\overbrace{a \times a \times \dotsb \times a}^{\text{\(m\) times}}\\ \times &a \times a \times \dotsb \times a\\ &\phantom{a \times a}\vdots\\ \times &a \times a \times \dotsb \times a\\ \vphantom{a} \end{align} %\right. \] So \(\left(a^m\right)^n\) is the same as multiplying \(n\) rows of \(m\) lots of \(a\), with the total of \(m \times n\) lots of \(a\) being multiplied together. This is the same as raising \(a\) to the power of \(m+n\).
12
Example Take \(13^4\) as an example. This is the same as \[13 \times 13 \times 13 \times 13.\] Now, raise that whole thing to the power of \(2\) and it'll become \begin{align} &(13 \times 13 \times 13 \times 13)\\ \times &(13 \times 13 \times 13 \times 13) = 13^{4 \times 2} \end{align} If instead of squaring, you cubed it: \begin{align} &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13 = 13^{4 \times 3} = 13^{12} \end{align} And, to the power of five: \begin{align} &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13 = 13^{4 \times 5} = 13^{20} \end{align}
Raising a Power of a Product Watch this space guys!
13
Be Careful! This does not work with addition!
Raising to Negative Powers Watch this space guys!
Raising to the Zeroth Power Watch this space guys!
Roots A root is just another fancy word for a solution, which means an answer to a problem. In this topic, we talk about the \(n\)th-roots as the solutions to the \(n\)th powers. Proceed to next page →
Square Roots We have a number, \(x\). The square roots of \(x\) are the numbers which, when squared, equals \(x\). This square root, call it \(y\), is the solution to \[y^2 = x.\] There's only one square root of zero (which is zero itself). There are no real square roots for negative numbers. There are two square roots for every other positive number. We write \(\sqrt{x}\) to represent the positive square root of \(x\). The other root would be \(-\sqrt{x}\). To write both possible roots, we use the plus or minus sign to get \(\pm\sqrt{x}\).
Cube Roots Cube roots of \(x\) are numbers which, when cubed, equals \(x\). We write them as \ (\sqrt[3]{x}\). If \(x\) is positive, \(\sqrt[3]{x}\) is positive. If \(x\) is negative, so as \(\sqrt[3]{x}\). 14
\(n\)th Roots Guess what? An \(n\)th root \(x\) is a number which when raised to the \(n\)th power, gives \(x\) back! Just like square roots and cube roots. We write the positive \(n\)th root of \(x\) as \(\sqrt[n]{x}\). It's nice to note that if \(n\) is even, then both \(\sqrt[n]{x}\) and \(-\sqrt[n]{x}\) are its \ (n\)th roots. If \(n\) is odd, then \(\sqrt[n]{x}\) is the only real root.
Surds Surds are irrational \(n\)th roots.
15
What do you mean by that? Well, consider \(\sqrt{4}\). This can be simplified down to an integer, \(2\), since \ (2^2=4\). Consider \(\sqrt{9}\), which simplifies to \(3\). What about \(\sqrt[3]{27}\)? That simplifies to \(3\) as well. Hence, all \(\sqrt{4}\), \(\sqrt{9}\) and \(\sqrt[3]{27}\) are rational \(n\)th roots. But what about \(\sqrt{2}\)? Can it be boiled down into a rational number? Turns out that it can't. There's a nice proof too. So, surds are \(n\)th roots that cannot be written as a fraction of two integers.
16
Example Please Square roots of: \(2\), \(3\), \(5\), \(6\), \(101\) and anything that is not a square of a rational number (hence irrational). These are surds. Cube roots of... well, of every integer that isn't cubes. These are also surds. And... so on. Did you notice that if the square root of an integer is not an integer, the square root is irrational? Pretty cool, huh?
Surds Can Be Multiplied In a nutshell: For all numbers \(a\) and \(b\), and integer \(n\), \[ \sqrt[n] {a\vphantom{b}\,} \sqrt[n]{\,\vphantom{a}b\,} = \sqrt[n]{ab}\,. \]
17
Explain, please Well, \begin{align} \left(\sqrt[n]{a\,}\right)^n &= a, &&\text{and}\\ \left(\sqrt[n] {\,b\,}\right)^n &= b, &&\text{by definition, so}\\ \left(\sqrt[n]{a\vphantom{b}\,} \sqrt[n]{\,\vphantom{a}b\,}\right)^n&=ab\\ \sqrt[n]{a\vphantom{b}\,} \sqrt[n] {\,\vphantom{a}b\,} &= \sqrt[n]{ab}\,. \end{align}
18
Explain more please [Sigh.] Okay.
Watch this precious space guys!
Surds Can Be Divided Watch this space guys!
Simplifying Expresions Including Surds Watch this space guys!
Rationalising the Denominator Watch this space guys!
Rationalising a Complicated Denominator Watch this space guys!
Raising to Fractional Powers Watch this space guys!
19
20
Chapter 2 Coordinate Geometry >
Cartesian Coordinate System Watch this space guys!
Distance Between Two Points Watch this space guys!
Straight Lines Watch this space guys!
21
Describing Straight Lines
Line Segments Watch this space guys! Midpoint of a Line Segment Watch this space guys!
Gradient of a Line Segment Watch this space guys!
The \(x\)-Intercept Watch this space guys!
The \(y\)-Intercept Watch this space guys!
The Equations of a Line Watch this space guys! ... although actually it should be:
22
The Line of an Equation ...but you know, it sounds weird, and it's important for it to sound good.
Equations of Vertical Lines Watch this space guys!
Equations of Horizontal Lines Watch this space guys!
Slope Intercept Form Watch this space guys!
Line With a Known Gradient Passing Through a Point Watch this space guys!
Line Passing Through Two Points Watch this space guys!
23
General Equation of a Straight Line Watch this space guys!
Finding the Gradient and the \(y\)Intercept of a Line Watch this space guys!
24
Intersection of Two Straight Lines Watch this space guys!
Gradients of Perpendicular Lines Watch this space guys!
25
Chapter 3 Displacement, Velocity and Acceleration (i.e. kinematics)
Watch this space guys!
Scalars and Vectors: Two Types of Quantities Watch this space guys!
Describing Motion: Kinematics Watch this space guys!
Displacement and Distances Watch this space guys!
Velocity and Speed Watch this space guys!
26
The Area Under a Graph Watch this space guys!
When Velocity is Constant Watch this space guys!
Acceleration and Deceleration Watch this space guys!
When Acceleration is Constant Watch this space guys!
Velocity, After a Certain Time Watch this space guys!
Displacement, After a Certain Time Watch this space guys!
Velocity, after a Certain Displacement 27
Velocity, after a Certain Displacement Watch this space guys!
Average Velocity During Constant Acceleration Watch this space guys!
Splitting a Scenario into Smaller Stages Watch this space guys!
28
Chapter 4 Representation of Data
29
Chapter 5 Measures of Location and Spread
30
Chapter 6 Quadratics
31
Chapter 7 Inequalities
32
Chapter 8 Functions
33
Chapter 9 Factorials
34
Chapter 10 Probability
35
Chapter 11 Permutations and Combination Permutations and combinations (a.k.a. perms and combs) is about counting possibilities. This topic can be tough, since exercises and questions need you to simplify the question into blocks of simple actions. Many intuitive yet incorrect working, and even worse, it's often very hard to check! Hang on tight ladies and gents, we'll march our way through.
Counting Perms and combs is all about counting. You count the number of ways certain things can happen, different ways you can do them, and the number of different possible results from what you do. For example, if there are three different T-shirts hanging within your wardrobe, there are three different ways of choosing a T-shirt from this wardrobe, and similarly the Tshirts could be arranged in six different ways along a line.
36
Processes The thing you do is usually called a process, or an operation. In the previous example the choosing of a T-shirt is a process, and arranging the T-shirts is also a process too. Rolling a die/dice, flipping a coin, and arranging people in a line are commonly processes used in perms and combs. What is an outcome? An outcome of a process is a result from doing the process. For example, getting a ``3'' from rolling a die, or choosing the first T-shirt, are both outcomes from their respective processes. The idea of processes and outcomes are also used in probability, as you will see later chapters. There are two main and easy rules to count. These will help you find and understand perms and combs. To easy? Carry on to the next page, and we'll see...
The Adding Rule (a.k.a. ``The fundamental principle of counting'') If you have two processes, and one of them can be done in \(x\) ways, while the other process can be done in \(y\) ways, then the total number of ways you can do one of the two processes would be the sum \(x+y\). This is obvious, as you will see: If flipping a coin gives you two possible outcomes (heads or tails), and rolling a die can give you six (numbers one to six), then there is a total of \(6+2=8\) different results from the ``flip a coin or roll a die'' process (heads, tails, and the numbers one to six).
37
Mutually Exclusive Processes Caution: I will now introduce a fancy term for something simple.
If there are two mutually exclusive processes, \(X\) and \(Y\), with \(x\) and \(y\) outcomes respectively, then there are \(x+y\) different outcomes in total for the process ``do \(X\) or \(Y\)''. Two processes are called mutually exclusive if they do not share a common outcome. This depends on what you take as an outcome: You can call the number on the top face of a die as an outcome e.g. five, or three You can also call the position where the die had landed as an outcome e.g. it landed here, or on top of the black square, etc. How you rolled the die, e.g. throwing or shaking it, can also be considered as the outcomes. Fliping a coin and rolling a die are mutually exclusive, as we consider the outcomes of rolling a die as different from heads or tails.
More Examples of Adding Numbers Walking along a street Imagine you are on a main street which splits to two smaller streets, one with three shops and the other with five shops. If all these shops were different, you would know that you have \(3+5=8\) different shops you can go to. Moreover, if some shops are connected to both smaller streets, you still know that you have \(3+5=8\) different choices you can make (i.e. pathways you can take followed by the shop you choose), even if the number of shops is less than eight. One important thing you do know is that you can't be on both streets at the same time, by common sense. From this you know that if there are \(3\) ways to go from street 1 and \(5\) ways to go from street 2, you have \(5+3=8\) ways to go, if you choose to go from ``street 1 or street 2''.
Adding Rule — Quiz Did you understant the most basic rule, the addition rule of counting? Let's see... (to be done)
The Multiplying Rule 38
You can multiply numbers in several cases. In general, if there are \(x\) additional things for every one of \(y\) other things, then there are \(xy\) altogether. What?! This is just like saying ``If each apple costs \($3\), and you need to buy five apples, the total cost would be \($3 \times 5 = $15\)''. Simple, if you have thirty additional people for every one class, and there are seven classes, you will have 210 people altogether. This easy, simple, yet powerful thinking is the key to many problems. Let's move on to the key examples of multiplication.
Repeated Addition Multiplication is just repeated addition. Remember that. Say you have \(x\) processes, from which each process have \(y\) different outcomes. If you need to choose one of the \(x\) processes and do that process, then from the adding rule, the number of possible outcomes is then \begin{align} &=\overbrace{y+y+y+\dotsb+y}^{\text{\(x\) times}}\\ &=xy. \end{align} For example, if you had five differently coloured fair dice, and you had to roll one of them, then you can get any one of the \(5 \times 6 \) outcomes if you take the colour of the die into account. For example the outcome could be (red,3). We can multiply because for every one of the \(x\) processes, there is an addition of \ (y\) outcomes. Hence there are \(x \cdot y\) total number of outcomes.
In a Multi-Step Process (a.k.a. ``The product rule of counting'') If you had to flip a coin, and then roll a 6-sided die, with the result of the coin and the die recorded down, you would have a total of \(6 \times 2 = 12\) number of possible outcomes. This is because you know that for every outcome of a coin, you have an addition of six more outcomes from the die. You also know that the outcome of the coin would surely not affect the outcome of the die (especially if you roll the die on another planet, for example). We call these processes as idenpendent. Two processes are independent if the outcome of one process doesn't affect the outcome of the other process. For two indepent processes, \(X\) and \(Y\), if they have \(x\) and \(y\) possible 39
outcomes respectively, then there are \(xy\) possible outcomes for the process ``do \ (X\) and \(Y\)''. This includes doing \(X\) first, then do \(Y\), or doing \(Y\) first, then do \(X\). The result will be the same set of possible outcomes, because \(X\) and \(Y\) are independent.
Avoid Duplicate Outcomes When working through a problem, a typical workflow might be something like: 1. Understand the problem. 2. Create a model of the problem. 3. Perform the calculations for the model. At step two, you take the scenario and subdivide it into small steps. After that, you figure out what to add and multiply from this model. Step two is basically figuring out what steps you would take to satisfy the problem, and then figuring out when you should add or multiply based on the model.
Example: Bob flips a coin, and then rolls a die. He records both of the results. How many possible results are there in total?
This can be broken down into two steps: 1. Flip a coin (two outcomes) 2. Roll a die (six outcomes) From this, we can use the multiplying rule (we don't need to state that explicitly though, since it's pretty obvoius). Two steps are independent, and since they are both done, the total number of outcomes is \(2\times 6 = 12\).
40
Caution Importantly, make sure: your method gives you all the possible outcomes (and doesn't miss any out), and your method avoids duplicate outcomes. If there are duplicate outcomes, you need to subtract them from the raw answer. If the model misses some of the possible outcomes, you need to add them back. This is like using the adding rule. In general, try to avoid duplicate outcomes, unless it is easier to subtract. Make sure you are absolutely certain in what you are doing, and not just doing it because it seems right.
Example: Bob tries to fit four people into a room (for some arbitrary reason), but only two people can be in it at once. The four people consists of two boys and two girls. How many different combination of people can he fit, if he fits two?
Arrangements Sometimes we like to arrange certain objects into some order. A permutation is a particular arrangement of objects. To permute objects means to rearrange them in some way. We are interested in the number of ways we can arrange a particular number of objects. Say there are \(n\) objects to order. There are several ways we can approach this. Read on to the next few pages to get the hang of permutations.
Take one, pass it along... Goal: We are trying to place \(n\) objects onto a table, and we take note of the order the objects are in. Assume that all objects are distinguishable from each other.
We can start with one object. We put it on a table. Now, we get another object. We put this on also on the table, next to the first object. We have two choices: we either put it to the left or to the right.
41
We then get our third object. We try to put this somewhere along the line of two objects. We have three choices: left end, right end, or in between. We continue, and for the \(k\)th object we have \(k\) choices: the \(k\) spaces to put the \ (k\)th object. We continue this until the end, the \(n\)th object, which will ultimately have \(n\) choices at which to fit the object in. We know that the number of choices we have at each object placing will be the same no matter what the previous choice was. Therefore we know that the number of possible outcomes at each step will be independent of the previous outcome (outcome as in the choice of the object's location). We can now see that for every possible placement of the second object, there are 3 choices for the third object, and for every possible arrangement of the first \(k\) objects, there are \(k+1\) choices for the \((k+1)\)th object. Therefore, the number of possible arrangements is equal to the product of each number of possible choices at each step, which gives us \[1 \times 2 \times 3 \times \dotsb \times n,\] which is the sum of all numbers from one to \(n\). This is the number of possible arrangements of \(n\) distinct objects.
Putting Objects into Boxes Goal: We are trying to fit \(n\) objects into \(n\) boxes and we take note of the order the objects are in. Assume that all objects are distinguishable from each other.
At the start, there are \(n\) objects waiting to be placed, and there are \(n\) boxes waiting for objects to be placed inside. Each box can only hold one object. We take one object, and we choose one box for it. There are \(n\) boxes to choose from. We put the object in, and now \(n-1\) objects are left to be placed in the remaining \(n-1\) boxes. We take a second object, and we choose one of the \(n-1\) boxes for it. There are \(n-2\) objects and boxes left. We continue this, and for the \(k\)th object, there are \(n-k\) boxes for it to choose from. Ultimately, the last object will have one remaining box left, leaving only one choice. We can see that every outcome of the box selection is independent of the previous arrangement. Hence, the number of possible arrangements is equal to \[n \times (n-1) \times (n-2) \times \dotsb \times 3 \times 2 \times 1.\] This is the same result as before, which is good!
Important Note 42
Question: Don't you need to also multiply by the number of ways you can select the objects in a order? E.g. for the first object, there are \(n\) possible choices for this object, and \(n\) possible choices for the box to put it in. Isn't it supposed to be \(n \times n \times (n-1) \times (n-1) \times (n-2) \times (n-2) \times \dotsb\) and so on?, because you've got \(k\) objects to choose on the \(k\)th object, and \(k\) boxes to put it in?? Answer: no. Why? Because... No matter what order you choose to pick the objects in, you can still get all the possible final arrangements. Otherwise, you get many duplicate arrangements. (See for yourself by drawing a tree diagram.) We can think of the objects as numbered already. Every object is distinguishable from each other. We are linking every object to different box.
Assigning Objects to each Box Goal: We are trying to assign \(n\) objects to every one of the \(n\) numbered boxes. Assume that all objects are distinguishable from each other.
We take the first box, and we try to find an object for it. There are \(n\) choices for this object, and we pick one, leaving \(n-1\) objects left unchosen. We take the second box, and we assign an object for it. There are \(n-1\) choices, and we leave \(n-2\) boxes left. You can see that this is going the same way as the other methods! We continue this until we reach the last box, where there is only one object left unchosen. All number of possible choices are unaffected by the previous choice, so we can multiply each of them at every step to get the total number of arrangements equal to \[n \times (n-1) \times (n-2) \times \dotsb \times 3 \times 2 \times 1.\]
Factorial Notation Fairly quickly you will find that writing out \(1 \times 2 \times \dotsb \times n\) as a whole will become boring and tedious, let alone pressing those silly buttons on a calculator to find the answer! That's when the factorial notation comes in! It even has its own calculator button. 43
In this notation, we write the phrase \[1 \\times 2 \\times 3 \\times \dotsb \\times n''\\] as the phrase \\[n!''\] Yep. It's that simple. Note that the exclamation mark does not mean I'm shouting it. The exlamation mark is actually part of the notation. The key symbol here is the exclamation mark, !'', after the number, pronounced here as\(n\)-factorial''. This would have the meaning of ``to multiply every single positive whole number less than and equal to itself''. For example, \(5! = 1 \times 2 \times 3 \times 4 \times 5\), which equals \(120\). You can see that the number of possible arrangements of \(n\) distinct objects is \(n!\) (that is, \(n\)-factorial).
b
Permutations Examples of Permutations Combinations Grouping Watch this space guys! Trigonometry
44
Chapter 15 Radians Radian is a way to measure angles. Before, you would write angles in degrees...
Now, you can also write angles in radians...
Don't get it? Don't worry, we'll go through it step by step.
Watch this space guys! Watch this space guys! Watch this space Geometric Series Vectors Vectors in 3D Three Components 45
Table of Contents Introduction
2
Surds & Indices
2
Types of Numbers Integers Rational Numbers Irrational Numbers Real Numbers Raising Powers Index Notation Multiplication Rule Division Rule Power-On-Power Rule Factor Rule Negative Powers Zero Power Roots
2 2 4 4 4 6 6 9 9 11 11 11 11 11
Square Roots Cube Roots nth Roots
11 11 11
Surds Multiplying Surds Dividing Surds Simplifying Surds
15 17 17 17
Rationalise Denominators Difference of Two Squares Fractional Indices
17 17 17
Coordinate Geometry
17
Cartesian Coordinates Distances
17 17
Straight Lines Line Segments Midpoint
17 17 17
Gradient x-Intercept y-Intercept Lines Have Equations Vertical Lines
17 17 17 17 17
Horizontal Lines
17 46
Slope Intercept Form
17
Point & Gradient Two Points General Equation
17 17 17
Finding Gradient & y-Intercept Intersection of Two Lines Perpendicular Lines
17 17 17
Velocity & Acceleration
17
Scalars & Vectors Describing Motion Displacement & Distance Velocity & Speed Area Under a Graph Constant Velocity Acceleration & Deceleration Constant Acceleration Velocity After Time
17 17 17 17 17 17 17 17 17
Displacement After Time Velocity After Displacement Average Velocity
17 17 17
Splitting Scenario into Stages
17
Representation of Data
17
Location & Spread Quadratics Inequalities
17 17 17
Functions Factorials Probability
17 17 17
Perms & Combs
17
Counting Adding Rule Adding Examples
17 17 17
Adding Quiz Multiplying Rule Repeated Addition Multi-step Processes Avoid Duplicate Outcomes Arrangements
17 17 17 17 17 17
47
Placing Object by Object Putting Objects into Boxes Assigning Objects to each Box Factorial Notation
41 41 41 41
Permutations Permutations Examples Combinations Grouping
41 41 41 41
Newton's Laws of Motion Differentiation Trigonometry
41 41 41
Trig are Ratios Unit Circle Period & Amplitude Sine Curve Cosine Curve Tangent Curve Common Values Periodic Properties
41 41 41 41 41 41 41 41
Translation Properties Solving Trig Identities
41 41 41
Radians
44
Using Radians Common Angles
44 44
Arc Length Sector Area Segment Area Trig Graphs
44 44 44 44
Trig Symmetry Inverse Trig
44 44
Friction
44
What's Happening? Coefficient of Friction
44 44
Arithmetic Series
44
Sequences Describing Sequences
44 44
Inductive Definition Triangle Numbers
44 44 48
Arithmetic Progressions Summing APs
44 44
Binomial Theorem
44
The Long Way Binomial Expansion Finding the Coefficients
44 44 44
Geometric Series
44
Geometric Sequece Summing GPs Converging Diverging Oscillating Exponential Growth & decay
44 44 44 44 44 44
Vectors
44
Vectors as Arrows Vectors aren't Arrows Column Vectors Scalars Multiplying by Scalars
44 44 44 44 44
Adding Vectors Subtracting Vectors Unit Vectors
44 44 44
Position Vectors Writing Vectors
44 44
Vectors in 3D
44
Three Components Base Vectors Adding, Subtracting, Multiplying No More Gradients
44 44 44 44
Skew Lines Planes Collinear Points
44 44 44
Magnitude Multiplying Magnitude Scalar Products
44 44 44
Sum of Components From Angle
44 44
What Does It Mean? Unit Vectors
44 44
49
Finding Angles Projections
44 44
Appendix – Useful Stuff
44
Sets Triangle Terms
44 44
50