Arithmetic

Definitions

Commutative property

A binary operation * on a set S is commutative if:

\forall x,y \in S (x * y = y * x)

We say that “x commutes with y“, or that “x and y commute under *“.

Anti-commutative property

A binary operation * on a set S is anti-commutative if:

\forall x,y \in S (x * y = -(y * x))

Associative property

A binary operation * on a set S is associative if:

\forall x,y,z \in S ((x * y) * z = x * (y * z))

Distributive property

Given a set S and two binary operations * and + on S, the operation * is distributive over the operation + if the operation * is both left-distributive and right-distributive over the operation +.

The operation * is left-distributive over the operation + if:

\forall x,y,z \in S (x * (y + z) = (x * y) + (x * z))

The operation * is right-distributive over the operation + if:

\forall x,y,z \in S ((y + z) * x = (y * x) + (z * x))

Closure property

A set S is closed under an n-ary operation \phi if:

\forall x_1,...,x_n \in S (\phi(x_1,..., x_n) \in S)

Identity element

An identity element of a set S with respect to a binary operation * is an element e such that:

\forall a \in S (a * e = e * a = a)

An element e \in S is called a left identity if:

\forall a \in S (e * a = a)

An element e \in S is called a right identity if:

\forall a \in S (a * e = a)

Factors

For a,b,c \in \mathbb{N}, the numbers a and b are called factors of c when c = a \cdot b.

Factors are transitive; if a is a factor of b, and b is a factor of c, then a is a factor of c.

Prime numbers

A number p \in \mathbb{N} is called a prime number if it has exactly two factors, 1 and p. 1 is not a prime number, as it has only one factor.

Fractions

A fraction is a number \frac{a}{b} such that \frac{a}{b} \cdot b = a.

The number above is called the numerator, and the number below is called the denominator.

Rational numbers

A number is a rational number if it can be expressed as a fraction of two integers with a non-zero denominator.

All integers are also rational numbers, as \forall a \in \mathbb{Z} (\frac{a}{1} = a).

Even and odd numbers

An even number a is an integer such that \exists b \in \mathbb{Z} (a = 2b).

An odd number a is an integer such that \exists b \in \mathbb{Z} (a = 2b + 1).

Operations

Addition

Addition is commutative and associative. \mathbb{N, Z, Q, R, C} are closed under addition. The operands of an addition operation are called the terms, addends, or summands, and the result is called the sum.

The additive identity is 0. The additive inverse of a number a is the number b = -a such that a + b = 0.

Subtraction

Subtraction is anti-commutative and non-associative. \mathbb{Z, Q, R, C} are closed under subtraction. The number being subtracted from is called the minuend, the number to subtract is called the subtrahend, and the result is called the difference.

Subtracting a number b from a is equivalent to adding the additive inverse of b to a:

\forall a,b (a - b = a + -b)

Multiplication

Multiplication is commutative and associative, and it distributes over addition and subtraction. The number to be multiplied is called the multiplicand, and the number to multiply by is called the multiplier, with both collectively being called factors. The result is called the product.

The multiplicative identity is 1. The multiplicative inverse of a number a is the number b = \frac{1}{a} such that a \cdot b = 1.

Division

Division is neither commutative nor associative. It is right-distributive over addition and subtraction. The number to be divided is called the dividend, the number to divide by is called the divisor, and the result is called the quotient. Division by 0 is undefined.

Dividing a number a by b is equivalent to multiplying a by the multiplicative inverse of b:

\forall a,b (\frac{a}{b} = a \cdot \frac{1}{b})

Theorems

Fundamental theorem of arithmetic

Any positive integer can be written as a unique product of primes. For example, 12 = 2^2 \cdot 3.

Proofs

Sum of an even and an odd number is odd

Given some even number x and some odd number y, show that the sum x + y is odd.

x + y = 2a + 2b + 1 for some integers a, b (via def. of odd and even numbers)
x + y = 2(a+b) + 1 (via distribution)
c = a + b for some integer c (via closure property of addition)
x + y = 2c + 1 (via substitution)

2c + 1 is in the form of an odd number, therefore the sum of an even and an odd number is odd.

Equivalent fractions

Given some integers a,b such that b \neq 0, show that:

\forall c \in \mathbb{Z} (\frac{a}{b} = \frac{a \cdot c}{b \cdot c})

Suppose \frac{a \cdot c}{b \cdot c} = q for some rational number q
a \cdot c = q \cdot (b \cdot c) (via multiplication by b \cdot c)
a \cdot c = (q \cdot b) \cdot c (via associativity)
a = q \cdot b (via division by c)
\frac{a}{b} = q (via def. of a fraction)

\frac{a}{b} = q and \frac{a \cdot c}{b \cdot c} = q, therefore the two fractions are equivalent \forall c \in \mathbb{Z}.

Multiplying two fractions

Given some integers a,b,x,y such that b,y \neq 0, show that:

\frac{a}{b} \cdot \frac{x}{y} = \frac{a \cdot x}{b \cdot y}

Suppose \frac{a}{b} = c, for some rational number c
Suppose \frac{x}{y} = z, for some rational number z
Suppose \frac{a \cdot x}{b \cdot y} = q for some rational number q
a \cdot x = q \cdot (b \cdot y) (via def. of a fraction)
(c \cdot b) \cdot (y \cdot z) = q \cdot (b \cdot y) (via substitution)
(b \cdot y) \cdot c \cdot z = q \cdot (b \cdot y) (via rearrangement)
c \cdot z = q (via division by b \cdot y)
\frac{a}{b} \cdot \frac{x}{y} = q (via substitution)

\frac{a \cdot x}{b \cdot y} = q and \frac{a}{b} \cdot \frac{x}{y} = q, therefore the two equations are equivalent.