Natural Numbers (\(\mathbb{N}\))

• Set of all positive and whole numbers excluding \(0\) 

e.g. - {\({1,2,3,4 ...}\)}

• Prime numbers are natural numbers that have exactly two factors.

• \(1\) is not a prime number.

• \(2\) is the only even prime number.

• Composite numbers are natural numbers greater than \(1\) which are not prime.

• Every natural number greater than \(1\) is either a prime or can be written as a unique product of primes.

• Factor or divisor of a natural number is any number that divides exactly into it.

• Multiples of a natural number are the numbers of which it divides into exactly.

• Highest Common Factor (HCF) of two or more natural numbers is the highest number that divides exactly into both given number.

• Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple that the given number has in common.

Examples of HCF and LCM

\(\text{e.g. 1 - Express 240 as a product of primes.}\)

              240
              / \
           (2)  120
                / \
              (2) 60
                  / \
                (2) 30
                    / \
                  (2) 15
                      / \
                    (3) (5)

\(240 = 2\times2\times2\times2\times3\times5\)

           \(=2^4\times3\times5\)

\(\text{e.g. 2 - Find (i) the HCF and (ii) the LCM of 512 and 280.}\)

\(512={2}\times{2}\times{2}\times2\times2\times2\times2\times2\times2\)             \(280={2}\times{2}\times{2}\times5\times7\)

\(512=2^9\)

\(\text{HCF}=2^3=8\)

\(\text{LCM}=2^3\times2^6\times5\times7=17920\)

Integers (\(\mathbb{z}\))

• Set of all whole numbers, positive, negative and \(0\).

e.g. - {\({... -3,-2,-1,0,1,2,3 ...}\)}

If \(a,b,c\) are integers:

• \(a+b\) and \(a \times b\) are integers

• Commutative property: \((a+b) = (b+a)\) and \(ab=ba\)

• Associative property: \((a+b)+c=a+(b+c)\) and \((ab)c=a(bc)\)

• Distributive property: \(a(b+c)=ab+ac\)

Rational Numbers (\(\mathbb{Q}\))

• Set of numbers that can be written as a ratio of two integers \({p}\over{q}\) where \(p,q\in\mathbb{z},q\neq0\)

• A rational number will have a decimal expansion that is either terminating or recurring.

• Any fraction whose denominator's prime factors are not only \(2\) and/or \(5\) is a recurring decimal.

• if \({p}\over{q}\) is a rational number, and \(p\neq0\), then \({q}\over{p}\) is its reciprocal.

Real Numbers (\(\mathbb{R}\))

• The set of real numbers is the set of all rational and irrational numbers.

Irrational Numbers (\(\mathbb{R}/\mathbb{Q}\)) or (\(\mathbb{I}\))

• Set of numbers that cannot be expressed as a ratio of two integers \({p}\over{q}\) where \(p,q\in\mathbb{z},q\neq0\)

• A surd (\(\sqrt{xyz}\)) is an irrational number containing a root term, i.e. \(\sqrt{2}=1.414213562....\), this number will continue indefinitely and has no pattern.

• \(\pi\) and \(e\) are two more examples of irrational numbers.

Note: \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\)

Proof that \(\sqrt2\) is irrational (Proof by Contradiction)

We will use proof by contradiction to prove: \(\sqrt2\) is irrational.

Proof: Assume that \(\sqrt2\) is rational and can therefore be written in the form of \({p}\over{q}\) where \(p\) and \(q\) have no common factors.

\(\sqrt2 = \frac{p}{q} \Rightarrow 2 = \frac{p^2}{q^2}\)

\(\therefore p^2 = 2q^2\)

\(\text{let }p=2k\)

\(\Rightarrow p^2=4k^2\)

\(p^2=2q^2\text{ and }p^2=4k^2\)

\(\Rightarrow 2q^2=4k^2\)

\(\Rightarrow q^2=2k^2\)

\(2\) is a factor of \(q^2\), so \(2\) is also a factor of \(q\)

Therefore \(2\) is a common factor of \(p\) and \(q\), this contradicts the original assumption. Thus \(\sqrt2\) is an irrational number.

Construct \(\sqrt2\)

Proof:

\(|ab|=|bc|=1\) (radii of circle)

\(|ab|^2+|bc|^2=|ac|^2\)

\(1^2+1^2=|ac|^2\)

\(2=|ac|^2\)

\(\sqrt2=|ac|\)

Construct \(\sqrt3\)

Proof:

\([CD]\) and \([AB]\) are perpendicular bisectors of each other.

\(\therefore |AE|=\frac{1}{2}|AB|=1\)

\(|AC|=1\) (Construction)

\(|AE|^2+|EC|^2=|AC|^2\)

\((\frac{1}{2})^2+|EC|^2=1^2\)

\(|EC|^2=1-\frac{1}{4}=\frac{3}{4}\)

\(\therefore |EC|=\sqrt\frac{3}{4}=\frac{\sqrt3}{2}\)

\(|CD|=2|EC|=2(\frac{\sqrt3}{2})\)

\(\therefore|CD|=\sqrt3\)

Rounding and Significant Figures

E.G. Write the following correct to one decimal place.

i) \(2.57\)

ans \(=2.6\)

ii) \(39.32\)

ans \(=39.3\)

E.G. Write the following numbers to two significant figures:

i) \(3.67765\)

ans \(=3.7\)

ii) \(61,343\)

ans \(=61000\)

iii) \(0.00356\)

ans \(=0.0036\)

E.G. Write the following numbers in scientific notation:

i) \(725,000,000,000\)

ans \(=7.25\times10^{11}\)

ii) \(980,000\)

ans \(=9.8\times10^{5}\)

iii) \(0.0000056\)

ans \(=5.6\times10^{-6}\)