---

---

Directed Number

$$3+4=7$$ $$3-4=-1$$ $$3+-4=3-4=-1$$ $$3--4=3+4=7$$ $$-3+4=1$$ $$-3+-4=-3-4=-7$$ $$-3\times -4=12$$

---

Order of Operation

The order in which operations are performed can be remember by the mnemonic,

BIDMAS

Brackets - Indices - Division and/or Multiplication - Addition and/or Subtraction

(Note: sometimes B for Brackets is replaced with P for Parentheses; sometimes I for indices is replaced with O for order)

This will help in remembering the correct priority of operations when completing calculations. Those on the left have priority over those on the right, so calculations are completed from left to right if BIDMAS doesn't help.

BODMAS (order of operations) Video:

Practice Questions;

Textbook Exercise.

(Care of corbettmaths.com)

BIDMAS:

Exam Questions;

The Order of Operations;

Solutions.

(Care of www.mathsgenie.co.uk)

---

Divisibility

An integer can be tested to see if it's divisibility using the following rules. Although you don't get an answer these are generally quicker than the complete calculation.

The ideas above have been used to show the divisibility of positive number by positive numbers. These ideas can be extended to include the negative numbers too.

---

Further Reading:

• Divisibility Rules
  (Care of www.mathsisfun.com)
• Check divisibility of 11
  (Care of www.wikihow.com)
• Divisibility rules for 13
  (Care of www.cuemath.com)
• Rules of Divisibility
  (Care of b28mathstutor.co.uk)
• Divisibility Rule
  (Care of en.wikipedia.org)
• Divisibility Rules
  (Care of www.goteachmaths.co.uk)

---

Follow-on Study

Prime Factor Decomposition

---

Ratio

To compare quantities, we can use ratios

Ratios can be simplified in the same way as fractions by finding common factors and cancelling.

To share out a quantity in a given ration, add the parts of the ratio to find the total number of parts, then divide the quantity by the number of parts to give the amount that each part represents. Then for each fraction of the ratio multiply the number of parts in each fraction by the quantity per part to give the amount for each fraction.

You can compare ratios by converting them into unitary form. This is where, at least, one of the fractions has a value of one and can be achieved by dividing all the fractions by the value of one of the fractions.

Ratio: simplifying Video

Practice Questions

Textbook Exercise

Ratio: express as fractions or % Video

Textbook Exercise

Ratio: sharing the total Video

Practice Questions

Textbook Exercise

Ratio: given one value Video

Practice Questions

Textbook Exercise

Ratio: given two ratios Video

Textbook Exercise

Ratio: difference between Video

Textbook Exercise

Ratio: expressing as 1:n Video

Textbook Exercise

Ratio: equations/ratios Video

Ratio: solving problems - Textbook Exercise [ MS ]

(All above care of corbettmaths.com)

---

Comparing Numbers

If $a,b\in ℝ$, then exactly one of the following is true:

If $a,b,c\in ℝ$, then:

$a<b$ and $b<c$ $\Rightarrow a<c$

If $a\in ℝ$, then there is $n\in \mathbb{N}$ such that:

$n>a$

---

Questions:

Ordering Decimals Video:

Practice Questions;

Textbook Exercise

PDF: ordering Video:

Practice Questions;

Textbook Exercise.

Ordering Fractions Video:

Practice Questions;

Textbook Exercise.

Ordering Negatives Video:

Practice Questions;

Textbook Exercise.

Ordering numbers Video:

Practice Questions;

Textbook Exercise

---

Combinations

If there are a ways to completing a task and b ways for a second task, then there are then $a\times b$ ways to do both tasks.

Allan, Bob and Charles enter a race
Write down the number of ways one can come first, another second and the last third.

Michael says that there are three ways for one of them to come first, only two for one to come second and a solitary way for third. Is michael correct?

How many different ways can these number of athelets finish?

1. 4 atheletes
2. 7 atheletes
3. 20 atheletes

Multiplying integers in decending (or ascending) order is call factorial
e.g., $4\times 3\times 2\times 1=4!$

---

Combinations
(care of justmaths.co.uk)

---

Rounding

Rounding to whole numbers Video:

Practice Questions;

Textbook Exercise.

Rounding to nearest 10 Video:

Practice Questions;

Textbook Exercise.

Rounding: to nearest 100 Video;

Practice Questions;

Textbook Exercise.

Rounding: to decimal places Video;

Practice Questions;

Textbook Exercise.

Rounding to Significant Figures Video:

Practice Questions;

Textbook Exercise;

Rounding to Significant Figures (Care of www.mathswithmum.com).

Rounding: highest/lowest Video;

Practice Questions;

Textbook Exercise

(All care of corbettmaths.com unless otherwise stated)

Rounding Numbers
(Care of www.mathsisfun.com)

---

Error Interval

Describing the error interval of a number:

The error interval can be thought of as the reverse process of rounding a number. A number given rounded to a known amount must accually lie between two numbers given by the rounded number, $n$, plus or minus half the rounding amount $\delta n$, i.e. $(n-\frac{\delta n}{2})\le n<(n+\frac{\delta n}{2})$

If $x$ the number $2$ has been rounded to the nearest the nearest whole number, then the error interval for $x=2$ is

$$1.5\le x<2.5$$

Note: $x$ is only less than $2.5$ as $2.5$ would round to $3$

---

Further Materials:

Error Intervals:

Practice Questions;

Textbook Exercise.

---