   ## Numbers (all)

Natural numbers     The numbers 1, 2, 3, 4, 5, … are called the natural numbers. They are also called the positive integers.

Cardinal and ordinal numbers     When natural numbers represent the quantity of something, they are called cardinal numbers. For instance, if we’re talking about 3 books, ‘3’ is a cardinal number. When they represent the order that something appears in, they are called ordinal numbers. For instance, if we’re talking about the 3rd book, ‘3rd’ is an ordinal number.

Integers     The integers consist of the positive integers, the negative integers and 0.

Integer part and fractional part     Given a real number a, the integer part of a, denoted by [a], is the largest integer not greater than a. a – [a] is called the fractional part of a, this is denoted by {a}. For example:

 = 3,   [3.2] = 3,   [-3.2] = -4

Divisibility     Let a, b be integers with b ≠ 0. If there exists an integer q such that a = q · b, we say that b divides a and write b | a. If no such q exists, we say that b does not divide a and write b ∤ a.

Factors (divisors) and multiples     If an integer a is divisible by an integer b (b ≠ 0), we say that a is a multiple of b and b is a factor of a. Every integer is a multiple of ±1 and ±1 is a factor of every integer. Since 0 is a multiple of every integer, every non-zero integer is a factor of 0.

Properties of the integers     Let a, b, c, d, … be integers

1. If a | b, then (-a) | b, a | (-b), (-a) | (-b), ∣a∣ | ∣b
2. If a | and b | c, then a | c
3. If a | b, then a | bc
4. If a | and c ≠ 0, then ac | bc
5. If ac | bc and c ≠ 0, then a | b
6. If a | b and b ≠ 0, then ∣a∣ < ∣b|
7. If ∣a∣ < ∣b∣ and ∣b∣ | ∣a∣, then a = 0
8. If d | a and d | b, then d | (a + b) and d | ab
9. If a_1 + a_2 + … + a_n = b_1 + b_2 + … + b_m, and among the (m + n) terms (m + n – 1) of them are multiples of d, then the remaining term is also a multiple of d.
10. Among m consecutive integers, at least one of them is divisible by m.
11. (Euclid’s division lemma) For any two integers a, b with b ≠ 0, there exist unique integers q, r such that a = qb + r, with 0 ≤ r < ∣b∣. a | b precisely when r = 0

Prime and composite numbers     For an integer greater than 1, if it’s only positive factors are 1 and itself, then it is called a prime number. If it has a positive factor different from 1 or itself, then it is called a composite number. 1 is neither prime nor composite.

Prime factors    If a positive integer a has a factor b and b is a prime number, we call b a prime factor of a. For example, 2 and 3 are the prime factors of 12. 4 and 6 are also factors of 12 but they are not prime.

Prime factorization     Writing a number as the product of all its prime factors is called forming the prime factorization of the number. For instance, the prime factorization of 12 is 12 = 2 × 2 × 3.

Odd numbers     Integers which are not divisible by 2 are called odd numbers. Every odd number is of the form 2k + 1 for some integer k.

Even numbers     Integers which are divisible by 2 are called even numbers.  0 is an even number. Every even number is of the form 2k for some integer k.

Common divisors and greatest common divisor     Let a_1, a_2, …, a_n (n ≥ 2) be n positive integers. If an integer d divides each one of them, i.e. d | a_1, a | a_2, … , d | a_n, then we call d a common divisor (or equivalently a common factor) of a_1, a_2, …, a_n. If d is the largest one among all the common divisors, we call it the greatest common divisor and write (a_1, a_2, …, a_n) = d to denote this. For instance, (18, 30, 66) = 6, since the common factors of 18, 30 and 66 are 1, 2, 3, 6.

Coprimality     Let a, b be integers. If (a, b) = 1, i.e. a and b have no positive divisors in common besides 1, then we say that a and b are coprime.

Common multiples and least common multiple     Let a_1, a_2, …, a_n (n ≥ 2) be n positive integers. If an integer m is a multiple for all of them, i.e. a_1 | m, a_2 | m, …, a_n | m, then we call m a common multiple of a_1, a_2, …, a_n. If m is the smallest among all the common multiples, we call m the least common multiple and write [a_1, a_2, …, a_n] = m to denote this.

Relationship between greatest common divisor and least common multiple     For any two positive integers a, b, we have (a, b) [a, b] = ab.

Rational numbers     Integers and fractional numbers are called rational number. Every rational number can be written in the form m / n for some integers m, n (n ≠ 0)

Irrational numbers     Numbers whose decimal expansions go on forever and don’t repeat are called irrational numbers.

Real numbers     Every real number is either rational or irrational.

Number line (?)

Explanation

Opposite numbers     A pair of numbers that differ only in sign are called opposite numbers. i.e. a and -a are a pair of opposite numbers. a is the opposite of -a and vice versa. 0 is the opposite of itself.

Absolute value     The absolute value of a positive real  and number is itself. The absolute value of a negative real number is its opposite. The absolute value of 0 is 0. Hence the absolute value is always non-negative, and is formed by ignoring any negative sign in front of the number. The absolute value denotes the distance of that number on the number line from the origin. Opposite numbers have the same absolute value.

Reciprocal     Given a non-zero real number a, the reciprocal of a is the real number b such that ab = 1. 0 does not have a reciprocal.

Comparing real numbers     For any two different numbers on the number line, the one on the right will be larger than the one on the left. The positive reals are all larger than 0 and the negative reals are all smaller than 0. Among two positive reals, the one with the larger absolute value is larger. Among two negative reals, the one with the larger absolute value is smaller.

Properties of the real numbers

1. For all real numbers a, we have a^2 ≥ 0, |a| ≥ 0. Equality holds precisely when a = 0.
2. If √a is a real number, then a ≥ 0.
3. The real numbers are ordered, for any pair of distinct real numbers, one will be greater than the other.
4. For any two real numbers a, b, we have Laws of addition     For any real numbers a, b, c, we have

(A) Commutativity      a + b = b + a

(B) Associativity    (a + b) + c = a + (b + c)

Laws of multiplication    For any real numbers a, b, c, we have

(A) Commutativity     ab = ba

(B) Associativity     (ab)c = a(bc)

(C) Distributivity     (a + b)c = ac + bc

Exponentiation     Multiplying n copies of the number a together is called exponentiation. We write . a is called the base, n is called the exponent and a^n is called the power.

Perfect powers     If b is the n-th power of a for positive integers a and n, then b is called a perfect power, or a perfect n-th power. In the case where b = a^2, b is called a perfect square.

Negative powers     a^-n is defined as 1/a^n

Roots     If a = b^n for some positive real numbers b and n, then b is the principal n-th root of a. We write . a is called the base and n is called the root exponent. When n = 2, b is called the square root of a. When n = 3, b is called the cube root of a. The n-th root of 0 is 0.

Note that exponentiation and taking roots are not inverse operations. For instance, 