The Integers #

The integers are a set that are comprised of $3$ subsets:

1. Positive Integers which is the set comprised of $\{1, 2, 3, 4, ... ,\}$
2. Natural Numbers which is the set comprised of $\mathbb{N} = \{0\} \cup \{1, 2, 3, 4, ...,\}$
3. Negative Integers which is the set comprised of $\{-1, -2, -3, -4, ...\}$

In summary:

$\begin{aligned} \mathbb{Z} &= \{-1,-2,-3,-4,...\} \cup \mathbb{N}\\ &= \{..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... \} \end{aligned}$

Operations: Addition & Subtraction #

Addition: Two Positive Integers #

Adding two positive integers results in a new positive integer, for example:

$\begin{aligned} 5 + 7 &= 12\\ 10 + 3 &= 13\\ 3 + 5 &= 8 \end{aligned}$

Observe the very simple rule for addition with $0$, namely:

$0 + a = a + 0 = a$

Meaning adding any integer ($a$) by $0$ yields that number ($a$).

Subtraction: Positive Integer + Negative Integer #

Adding a positive integer with a negative integer has the following rules:

For any positive integer $a$ and any negative integer $b$:

1. If $a > b$ then $a + b$ is a positive integer.
2. If $a = b$ then $a + b$ is $0$.
3. If $a < b$ then $a + b$ is a negative integer.

Examples:

$\begin{aligned} 10 + (-5) &= 5\\ 3 + (-3) &= 0 \\ 10 + (-12) &= -2 \end{aligned}$

Adding by a negative number is also called subtraction and is commonly denoted using the following expression: $a - b$ where $a, b$ are positive integers.

The previous example can then be described as follows:

$\begin{aligned} 10 - 5 &= 5\\ 3 - 3 &= 0 \\ 10 - 12 &= -2 \end{aligned}$

Subtraction: Negative Integers + Negative Integers #

Subtracting two negative integers always results in a larger negative integer.

Examples:

$\begin{aligned} -1 - 1 &= -2\\ -8 - 7 &= -15\\ -10 + (-18) &= -28 \end{aligned}$

Rules for addition #

Commutativity #

If $a, b$ are integers then:

$a + b = b + a$

For example:

$\begin{aligned} 5 + 3 = 3 + 5 &= 8\\ -2 + 5 = 5 - 2 &= 3\\ -3 - 4 = -4 - 3 &= -7 \end{aligned}$

Associativity #

If $a, b, c$ are integers, then:

$(a + b) + c = a + (b + c)$

For example:

$\begin{aligned} (3 + 5) + 9 = 8 + 9 &= 17\\ 3 + (5 + 9) = 3 + 14 &= 17 \end{aligned}$

Associativity also works for negative integers:

$\begin{aligned} (-2 - 3) - 4 = -5 - 4 &= -9\\ -2 + (-3 - 4) = -2 - 7 &= -9 \end{aligned}$

Sign Inversion #

If $a + b = 0$, then $b = -a$ and $a = -b$.

To prove this:

1. Subtract both sides by $-a$: $-a + a + b = 0 - a$, which yields to $b = -a$. This proves the first part.
2. To prove the second part, invert $b$ to its negative form, then we get $-b = -(-a)$ which yields to $-b = a$, proving the second part of the statement.

The expression $a = -(-a)$ is true because $a + (-a) = 0$. Applying $a + b = 0$ where $b = -a$ this means:

1. If $b$ is negative, then $-a$ is positive (which means $-(-a)$ is positive ie. $-b = -(-a)$).
2. If $b$ is positive, then $-a$ is negative.

As a consequence:

$-(a + b) = -a + (-b)$

or

$-(a + b) = -a - b$

Proof

Remember that if $x, y$ are integers, then $x = -y$ and $y = -x$ means that $x + y = 0$. To prove the assertion that $-(a + b) = -a - b$, we must show that:

$(a + b) + (-a - b) = 0$

Where $x = -y = (a + b)$ and $y = -x = (-a - b)$.

This is done by the ff.

$\begin{aligned} (a + b) + (-a - b) &= a + b - a - b\\ &= a - a + b - b\\ &= 0 + 0\\ &= 0 \end{aligned}$

Which proves the formula.

Examples:

$\begin{aligned} -(3 + 5) = -3 - 5 &= -8\\ -(-2 + 3) = -(-2) - 3 = 2 - 3 &= -1\\ -(3 - 7) = -3 - (-7) = -3 + 7 &= 4 \end{aligned}$

Law of Cancellation in Addition #

If we have the relationship between three numbers:

$a + b = c$

then we can derive other relationships between them. For instance, adding $-b$ on both sides of this equation we get:

$\begin{aligned} a + b - b &= c - b\\ a + 0 &= c - b \end{aligned}$

Similarly we can conclude that,

$\begin{aligned} a - a + b &= c - a\\ 0 + b &= c - a \end{aligned}$

For instance, if

$x + 3 = 5$

then

$\begin{aligned} x + 3 &= 5\\ x + 3 - 3 &= 5 - 3\\ x + 0 &= 2 \\ x &= 2 \end{aligned}$

Operations: Multiplication #

Multiplication means adding a number to itself several times.

Let $a, b$ be some numbers, then their multiplication would be:

$a \cdot b$

or simply denoted as just:

$ab$

where $a$ is added to itself $b$ number of times.

For example:

$\begin{aligned} 4 + 4 = 4 \cdot 2 &= 8\\ 2 + 2 + 2 = 2 \cdot 3 &= 6\\ 13 + 13 + 13 + 13 = 13 \cdot 4 &= 52 \end{aligned}$

Rules for Multiplication #

For any integer $a$, the rules of multiplying by $1$ and $0$ are:

$\begin{aligned} 1a = a\\ 0a = 0 \end{aligned}$

Let $a, b, c$ be some number, then these properties follow:

Commutativity #

$ab = ba$

Associativity #

$(ab)c = a(bc)$

Using these properties we can now do something which is often useful: multiplying constants.

For example:

$\begin{aligned} (2a)(3b) &= 2(a(3b))\\ &= 2(3a)b\\ &= (2\cdot3)ab\\ &= 6ab \end{aligned}$

Distributivity #

$a(b + c) = ab + ac$

Inverse #

$(-1)a = -a$

$-(ab) = (-a)b = a(-b)$

Two Negatives make a Positive #

$(-a)(-b) = ab$

Powers #

Multiplying a number with itself several times is called getting the power of that number.

For example:

$\begin{aligned} aa &= a^{2}\\ aaa &= a^{3}\\ aaaaa &= a^{4} \end{aligned}$

Or in general, given some number $a$ and a positive integer $n$:

$a^{n} = aaa...a$

Rules for Powers #

If $m, n$ are positive integers, then:

Adding Powers #

$a^{m + n} = a^{m}a^{n}$

For example:

$\begin{aligned} a^{2}a^{3} &= (aa)(aaa) = a^{2 + 3} = aaaaa = a^{5}\\ (4x)^{2} &= 4x \cdot 4x = 4\cdot4xx = 16x^{2}\\ (7x)(2x)(5x) &= 7\cdot2\cdot5xxx = 70x^{3} \end{aligned}$

Multiplying Powers #

$(a^{m})^{n} = a^{mn}$

The following three formulas are used constantly. They are so important that they should be thoroughly memorized!

$(a + b)^{2} = a^{2} + 2ab + b^{2}$

$(a - b)^{2} = a^{2} - 2ab + b^{2}$

$(a + b)(a - b) = a^{2} - b^{2}$

Rational Numbers #

Given two numbers $m, n$ where $n \neq 0$. A rational number is the fraction of these two numbers:

$\frac{m}{n}$

For example:

$\frac{1}{4}, \frac{2}{5}, \frac{7}{3}$

Rule for Cross-Multiplying #

Let $m, n, r, s$ be integers and assume that $n \neq 0$ and $s \neq 0$. Then:

$\frac{m}{n} = \frac{r}{s} \Leftrightarrow ms = rn$

For example:

$\frac{1}{2} = \frac{2}{4}$

because

$1\cdot4 = 2\cdot2$

We shall make no distinction between an integer $m$ and the rational number $\frac{m}{1}$. Therefore:

$m = \frac{m}{1}$

Cancellation Rule for Fractions #

Let $a$ be a non-zero integer. Let $m, n$ be integers, where $n \neq 0$. Then:

$\frac{am}{an} = \frac{m}{n}$

Also observe that:

$\frac{-m}{n} = \frac{m}{-n}$

via this proof which uses the cross-multiplying rule:

$(-m)(-n) = mn$

Divisibility #

The cancellation rule corrolary provides the notion of divisibility where given some integer $d$, it is the common divisor of $m, n$.

$\frac{m}{n} = \frac{dm}{dn} = \frac{r}{s}$

For example:

$\frac{2}{3} = \frac{5\cdot2}{5\cdot3} = \frac{10}{15}$

Common Denominator #

Let $\frac{m}{n}$ and $\frac{r}{s}$ be rational numbers, expressed as quotients of integers. We can put these rational numbers over a common denominator $ns$ by writing.

$\frac{m}{n} = \frac{ms}{ns}$

and

$\frac{r}{s} = \frac{nr}{ns}$

For example:

We can put $\frac{3}{5}$ and $\frac{5}{7}$ over the common denominator $5\cdot7 = 35$, we write:

$\frac{3}{5} = \frac{3\cdot7}{5\cdot7} = \frac{21}{35}$

and

$\frac{5}{7} = \frac{5\cdot5}{7\cdot5} = \frac{25}{35}$

This leads us to the formula for adding rational numbers:

$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}$

For example:

$\frac{-3}{8} + \frac{2}{8} = \frac{-3 + 2}{8} = \frac{-1}{8}$

When the rational numbers do not have a common denominator then we can get the formula by each side via their respective denominators ie.

$\frac{m}{n} = \frac{sm}{sn}$

and

$\frac{r}{s} = \frac{nr}{ns}$

For example:

$\frac{3}{5} + \frac{5}{7} = \frac{3\cdot7}{5\cdot7} + \frac{5\cdot5}{7\cdot5} = \frac{21}{35} + \frac{25}{35} = \frac{21 + 25}{35} = \frac{46}{35}$

Observe that dividing by $0$ has the property:

$\frac{0}{n} = 0$

For any integer $n \neq 0$.

Multiplying Rational Numbers #

Multiplying rational numbers simply involves multiplying their numerators and multiplying their denominators respectively.

$\frac{m}{n} \cdot \frac{r}{s} = \frac{mn}{rs}$

For example:

$\frac{3}{5} \cdot \frac{7}{8} = \frac{3\cdot7}{5\cdot8} = \frac{21}{40}$

Powers should work the same:

$(\frac{r}{s})^{n} = \frac{r^{n}}{s^{n}}$

For example:

$(\frac{2}{5})^{3} = \frac{2^{3}}{5^{3}} = \frac{8}{125}$

Multiplicative Inverses #

Rational numbers satisfy one property which is not satisfied by integers, namely:

If $a$ is a rational number $\neq 0$, then there exists a rational number denoted by, $a^{-1}$, such that

$a^{-1}a = aa^{-1} = 1$

Indeed, if $a = \frac{m}{n}$ where $m, n$ are integers and $n \neq 0$, then $a^{-1} = \frac{n}{m}$ because:

$\frac{m}{n} \cdot \frac{n}{m} = \frac{mn}{mn} = 1$

We call $a^{-1}$ the multiplicative inverse of $a$.

Example:

The multiplicative inverse of $\frac{1}{2} = \frac{2}{1} = 2$ because:

$2\cdot\frac{1}{2} = 1$

The multiplicative inverse of $\frac{2}{3} = \frac{3}{2}$ and the multiplicative inverse of $-\frac{5}{7} = -\frac{7}{5}$.

Observe that if $a$ and $b$ are rational numbers such that:

$ab = 1$

then

$b = a^{-1}$

Proof: We multiply both sides of the relation $ab = 1$ by $a^{-1}$ and get:

$a^{-1}ab = a^{-1}\cdot1 = a^{-1}$

Which means

$b = a^{-1}$

This rule allows us to divide fractions with other fractions, for example:

$\frac{\frac{3}{4}}{\frac{5}{7}} = \frac{3}{4}(\frac{5}{7})^{-1} = \frac{3}{4}\cdot\frac{7}{5} = \frac{21}{20}$

Cross-multiplication #

Let $a, b, c, d$ be rational numbers where $b,d \neq 0$.

$\frac{a}{b} = \frac{c}{d} \rightarrow ad = bc$

$ad = bc \rightarrow \frac{a}{b} = \frac{c}{d}$