Number

Types of numbers

Natural numbers

• All naturally occurring numbers: the positive whole numbers.
• 0 is not included.
• $1, 2, 3, 4, 5, 6, \dots$

Integers

• All positive and negative whole numbers.
• 0 is included.
• $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

Prime numbers

• Numbers that can only be evenly divided by themselves and 1.
• In short, a prime number only has 2 factors: itself and 1.
• 1 is not a prime number.
• $2, 3, 5, 7, 11, \dots$

Square numbers

• A square number is a number you get when you multiply a number by itself.
• For example, 4 (because $2\times2$), 9 (because $3\times3$), 16, 25, 36, and so on.

Cube numbers

• A cube number is a number you get when you multiply a number by itself 2 times (three of the number are multiplied).
• For example, 8 (because $2\times2\times2$), 27 (because $3\times3\times3$), and so on.

Rational and irrational numbers

• A rational number is any number whose decimal stops or repeats.
• It is a number that can be written as a fraction.
• For example, $2$, $3.1$, $\frac{1}{3}$ ($0.33333\dots$), $-\frac{3}{5}$.
• An irrational number cannot be written as a fraction.
• Its decimal goes on forever and never repeats.
• For example, $\pi$ ($3.14159265\dots$), $\sqrt{2}$.

Highest common factor (HCF)

How to find the HCF:

1. Prime factorise each number.
2. Choose the lowest power of each common factor.
3. Multiply them.

$24 = 2^3 \times 3^1$ and $36 = 2^2 \times 3^2$.

Smallest powers: $2^2$ and $3^1$. Hence HCF $= 2^2 \times 3 = 12$.

Divide all three by a number that goes into all of them, until none does:

HCF $= 2\times2\times3 = 12$.

Lowest common multiple (LCM)

How to find the LCM:

1. List the prime factors of each number.
2. Choose the highest power of all prime factors (they don't need to be common factors).
3. Multiply them together.

• $24 = 2^3 \times 3^1$
• $36 = 2^2 \times 3^2$
• Highest powers: $2^3$ and $3^2$
• Hence LCM $= 2^3 \times 3^2 = 72$

Reciprocals

• A reciprocal is what you get when you flip a number upside down, especially a fraction.
• For example, $4 \to \frac{1}{4}$ and $\frac{3}{5} \to \frac{5}{3}$.

Practice questions

Rational: $-2$, $0.75$, $\frac{1}{3}$ and $0.666\dots$. Not rational: $\sqrt{2}$ and $\pi$.

Start dividing 96 by the smallest prime number and keep dividing by prime numbers until it can no longer be divided further:

• $96 \div 2 = 48$
• $48 \div 2 = 24$
• $24 \div 2 = 12$
• $12 \div 2 = 6$
• $6 \div 2 = 3$

So $96 = 2\times2\times2\times2\times2\times3 = 2^5 \times 3$.

A harder HCF and LCM question

(a) Divide by the smallest prime numbers: $108 \div 2 = 54$, $54 \div 2 = 27$, $27 \div 3 = 9$, $9 \div 3 = 3$, $3 \div 3 = 1$. So $108 = 2^2 \times 3^3$.

(b) The LCM takes the highest powers of all primes involved.

• $108 = 2^2 \times 3^3$
• $360 = 2^3 \times 3^2 \times 5$
• Highest power of 2: $2^3$. Highest power of 3: $3^3$. 5 appears only in 360, so take 5.

Hence LCM $= 2^3 \times 3^3 \times 5$.

(c) We need $k$ so that $360 \times k$ is a perfect cube. For a cube number, all prime powers must be multiples of 3.

• $2^3$: already a multiple of 3.
• $3^2$: needs 1 more 3 to become $3^3$.
• $5$: needs 2 more 5s to become $5^3$.

So we must multiply 360 by $3 \times 5^2 = 3 \times 25 = 75$. The smallest value of $k$ is 75.

To find the smallest $n$ so that $225 \times n$ is a multiple of 540, $225n$ must have all the prime factors of 540, with the same or bigger powers of each prime.

• 225 has no 2s at all, but 540 has two 2s ($2^2$), so we need to multiply by $2^2 = 4$.
• 225 has $3^2$, but 540 has $3^3$, so we are missing one more 3. Multiply by 3.
• 225 already has enough 5s (it has $5^2$ and 540 only has one 5).

Multiply all the missing pieces together: $4 \times 3 = 12$. The smallest value of $n$ is 12.

Sets

A set is a collection of items (called elements or members) that are grouped together because they have something in common.

• We usually write sets using curly brackets $\{\ \}$, and separate each element with a comma.
• Example: set $A = \{2, 4, 6, 8\}$ is a set of even numbers.

Set builder notation

Instead of listing every element, we can describe the rule: $\{x : \text{condition about } x\}$ means "the set of all $x$ such that $x$ follows a certain rule".

• $A = \{x : x \text{ is a natural number}\}$: $A$ is the set of all natural numbers.
• $B = \{(x, y) : y = mx + c\}$: $B$ is the set of points that lie on the line $y = mx + c$.
• $C = \{x : a \le x \le b\}$: $C$ is the set of numbers between $a$ and $b$, including $a$ and $b$.
• $D = \{a, b, c, \dots\}$: $D$ is a set containing the elements $a, b, c, \dots$

Venn diagrams

• Venn diagrams are pictures that show how sets relate to each other.
• They use circles inside a rectangle (which represents the universal set).

Powers and roots

Squares and cubes have been introduced before.

The square root is the opposite of squaring. It means: what number multiplied by itself gives this value?

Cube root means: what number multiplied by itself 3 times gives this? For example, $\sqrt[3]{64} = 4$ because $64 = 4\times4\times4$.

You can also work with:

• Higher powers like $2^4 = 2\times2\times2\times2 = 16$.
• Higher roots like $\sqrt[4]{81} = 3$ because $3\times3\times3\times3 = 81$.

Fractions, decimals and percentages

A decimal is another way of writing part of a whole number using a decimal point. For example, $\frac{1}{2} = 0.5$ and $\frac{3}{4} = 0.75$.

Recurring decimals

Some decimals repeat forever. The notation puts a dot or bar over the repeating digits.

• $0.1\dot{7} = 0.17777\dots$
• $0.\overline{123} = 0.123123123\dots$
• $0.1\dot{2}\dot{3} = 0.1232323\dots$

Converting percentages

Percent means "per 100". Hence 40% means 40 for every 100. The percentage symbol just means divided by 100.

Ordering and BODMAS

BODMAS

• Brackets
• Orders (powers and roots)
• Division
• Multiplication
• Addition
• Subtraction

Follow this order while simplifying long expressions (left to right).

$(6+4)\div2^2 + 3\times(5-2) \to 10 \div 2^2 + 3\times3 \to 10\div4 + 3\times3 \to 2.5 + 9 = 11.5$

Adding and subtracting fractions

First, let's learn how to convert the different types of fractions into other forms.

Mixed to improper. Example: $2\frac{1}{3}$.

1. Multiply the whole number by the denominator: $2\times3 = 6$.
2. Add the numerator: $6 + 1 = 7$.
3. Keep the same denominator. Answer $= \frac{7}{3}$.

Improper to mixed. Example: $\frac{9}{4}$.

1. Divide the top by the bottom: $9 \div 4 = 2$ with remainder 1.
2. Write the remainder as the numerator. Answer: $2\frac{1}{4}$.

Make the denominators the same by multiplying each fraction by the other's denominator:

$\frac{9\times5}{4\times5} + \frac{3\times4}{5\times4} = \frac{45}{20} + \frac{12}{20} = \frac{57}{20}$

Use the same technique for subtraction as well.

Indices

An index (plural: indices) tells you how many times to multiply a number by itself. Example: $5^3 = 5\times5\times5 = 125$. Here, 5 is the base, and 3 is the index or power.

Types of indices you should know

• Positive indices mean regular multiplying: $2^4 = 2\times2\times2\times2 = 16$.
• Zero index: any number to the power of 0 is always 1: $7^0 = 1$.
• Negative indices: flip the number (take the reciprocal): $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

$27^{-1/3} = \frac{1}{27^{1/3}} = \frac{1}{\sqrt[3]{27}} = \frac{1}{3}$

Use the same base rule: add the powers if multiplied, subtract if divided. Since we are multiplying, we add the powers:

$2^{-2+5} = 2^3 = 8$

Work inside the brackets first (BODMAS), using $\frac{x^3}{x} = x^{3-1}$ and $\frac{y}{y^3} = y^{1-3}$:

$\left(3x^2y^{-2}\right)^{-2} = 3^{-2}\,x^{2\times(-2)}\,y^{(-2)\times(-2)} = \frac{1}{3^2}\,x^{-4}\,y^{4} = \frac{y^4}{9x^4}$

Write 16 as a power of 4: $(4^2)^b = 4$, so $4^{2b} = 4^1$. For the same base, we can take the powers as a separate equation: $2b = 1$, so $b = \frac{1}{2}$.

Standard form

Standard form is also called scientific notation. It helps us write very big or very small numbers in a simple way, using powers of 10. It looks like this:

$a \times 10^n$

where $a$ is a number between 1 and 10 ($1 \le a < 10$) and $n$ is an integer (it can be positive or negative).

How to convert a big number to standard form

1. Move the decimal point so there is only one digit before it.
2. Count how many places you moved it. That is the power of 10.

Example: $53000 \to 5.3\times10^4$ (the decimal point moved four places to the left).

How to convert a small number to standard form

1. Move the decimal point after the first non-zero digit.
2. Count the places moved. This time the power of 10 is negative.

Example: $0.0076 \to 7.6\times10^{-3}$ (the decimal point moved three places to the right).

Estimation

Estimation helps us get a rough idea of an answer without needing to work it out exactly. It is useful when we want to save time and check if an answer is sensible, or when an exact value is not needed.

You may be asked to round:

• to the nearest 10, 100, 1000 (place value)
• to a certain number of decimal places
• to a certain number of significant figures

Significant figures are digits that carry meaning.

Round 5764 to:

• the nearest thousand: 6000
• 1 significant figure: 6000
• 2 significant figures: 5800

Round 0.004678 to:

• 1 significant figure: 0.005
• 2 significant figures: 0.0047

Step 1. Round each number to 1 significant figure: $41.3 \approx 40$, $9.79 \approx 10$, $0.765 \approx 0.8$.

Step 2. Estimate: $\dfrac{40}{10\times0.8} = \dfrac{40}{8} = 5$.

Limits of accuracy

• When we round a number, the real value could be slightly more or less than the rounded number.
• So we talk about the smallest and largest the actual value could be.
• These are called the lower bound and the upper bound.

Example. The number is 72, rounded to the nearest whole number. Nearest whole number means we round to the nearest 1. Lower bound $= 71.5$ and upper bound $= 72.5$.

When numbers are used in calculations, we can calculate the maximum and minimum possible answers using the bounds.

• Length upper bound $= 8.45$
• Width upper bound $= 5.15$
• Maximum area $= 8.45 \times 5.15 = 43.5175\ \text{cm}^2$

Ratio and proportion

A ratio compares two or more values. It shows how much of one thing there is compared to another.

Simplifying ratios

Example: $20 : 30 : 40$. Find the HCF (highest common factor), which is 10, and divide each part by 10 to get $2 : 3 : 4$.

Dividing a quantity in a ratio

Example: divide 90 in the ratio $2 : 3$.

• Step 1: add the parts: $2 + 3 = 5$.
• Step 2: divide 90 by 5: $90 \div 5 = 18$.
• Step 3: multiply: first part $2\times18 = 36$ and second part $3\times18 = 54$.

Answer: 36 and 54.

Using ratios in real life

Flour : sugar is $3 : 2 = 9 : x$. Cross-multiplying, $3x = 18$, so $x = 6$ cups of sugar.

That means 1 cm on the map is 1000 cm in real life. So 5 cm on the map is $5 \times 1000 = 5000$ cm.

Rates

A rate tells us how much something happens over a certain time or space. We usually see it written like km/h (kilometres per hour) or g/cm³ (grams per cubic centimetre).

Hourly rate $= \frac{72}{6} = 12$ dollars per hour.

Density $= \frac{\text{mass}}{\text{volume}} = \frac{200}{100} = 2\ \text{g/cm}^3$.

3 hours 45 minutes $= 3.75$ hours. Speed $= \frac{\text{distance}}{\text{time}} = \frac{45}{3.75} = 12$ km/h.

For questions about speed, distance and time, remember this triangle, with distance (D) on top and speed (S) and time (T) underneath:

• To find distance, multiply speed and time: $D = S \times T$.
• To find speed, divide distance by time: $S = \frac{D}{T}$.
• To find time, divide distance by speed: $T = \frac{D}{S}$.

Percentages

Calculate a given percentage. Find 30% of 80: $\frac{30}{100}\times80 = 24$.

Express one quantity as a percentage of another. What percentage is 20 out of 50? $\frac{20}{50}\times100 = 40\%$.

Percentage increase and decrease

Increase $= 50\times\frac{10}{100} = 5$ dollars. Final price $= 50 + 5 = 55$ dollars.

Discount $= \frac{20}{100}\times200 = 40$ dollars. Final price $= 200 - 40 = 160$ dollars.

Simple and compound interest

Simple interest

$\text{Interest} = \frac{P \times R \times T}{100}$

where $P$ is the principal (starting money), $R$ is the rate (%) and $T$ is the time (years).

Interest $= \frac{1000\times5\times2}{100} = 100$ dollars. Total money available $=$ interest $+$ principal $= 100 + 1000 = 1100$ dollars.

Compound interest

$F = P\left(1 + \frac{r}{100}\right)^n$

where $F$ is the final money available, $P$ is the principal, $r$ is the rate and $n$ is the number of terms (if the compounding is yearly, $n$ is the number of years).

$F = 100\left(1 + \frac{10}{100}\right)^2 = 121\ \text{dollars}$

In the simple interest question above, there is a total interest of $100 over 2 years, so the same amount ($50) is added each year.

• Compound interest means you earn interest not just on your original money, but also on the interest you already earned.
• So in this question, someone puts $100 in the bank, and the bank gives 10% interest every year.
• After the first year, 10% of $100 is $10, so the total becomes $110. In the second year, the 10% is now calculated on $110, which gives $11 more. That makes the total after two years $121.

So the amount keeps growing faster, because each year the interest is added to the new total, not just the starting amount.

Reverse percentages

In the previous questions, we calculated the change and then added or subtracted it from the initial cost. Using this idea, we get:

$\text{Initial cost} \times \frac{100 + \text{percentage change}}{100} = \text{Final price}$

So:

$\text{Cost price} = \frac{120}{(20+100)\%} = \frac{120}{1.2} = 100\ \text{dollars}$

If the question said that the final price was after a 20% decrease, the percentage change would be negative, which gives:

$\text{Cost price} = \frac{120}{(-20+100)\%} = \frac{120}{0.8} = 150\ \text{dollars}$

Money

Use the ratio method. Euro : PKR is $1 : 310 = 7 : x$, so $x = 7\times310 = \text{Rs. } 2170$.

Surds

A surd is a square root (or other root) that can't be simplified into a whole number. It just keeps going and going, like a never-ending decimal. Think of $\sqrt{2}$ or $\sqrt{3}$: you can't write them as exact decimals.

Simplifying surds

We simplify surds by finding square numbers inside them.

• $\sqrt{20} = \sqrt{4\times5} = \sqrt{4}\times\sqrt{5} = 2\sqrt{5}$
• $\sqrt{200} = \sqrt{100\times2} = \sqrt{100}\times\sqrt{2} = 10\sqrt{2}$

We can only add surds that have the same root.

• $\sqrt{3} + \sqrt{5}$: not possible to combine (the roots are different, so the answer stays as $\sqrt{3} + \sqrt{5}$).
• $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$
• $\sqrt{8} + \sqrt{2} = \sqrt{2\times2\times2} + \sqrt{2} = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$

Rationalising the denominator

Multiply top and bottom by the conjugate, which is $(1 - \sqrt{3})$:

$\frac{1}{1+\sqrt{3}}\times\frac{1-\sqrt{3}}{1-\sqrt{3}} = \frac{1-\sqrt{3}}{(1+\sqrt{3})(1-\sqrt{3})} = \frac{1-\sqrt{3}}{1-3} = \frac{1-\sqrt{3}}{-2} = \frac{\sqrt{3}-1}{2}$

$\frac{10}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}$

For normal denominators, multiply the surd in the denominator with the numerator and the denominator.