Zero is neither positive nor negative.
Any positive integer (natural number) and the negative integer obtained by negating it are called counterpart integers. For example, +4, -4, +10, -10, 7, -7 are pairs of counterpart integers. On the number line, zero is the midpoint of the line segment joining points located by any pair of counterpart numbers.
[1.1] (Absolute value The absolute value of either of a pair of counterpart numbers is the same and is equal to the number that is not negative.
Notation: |+4| = |-4| = 4, |-10| = |+10| = 10, |7| = |-7| = 7, |0| = 0
The absolute value is never negative. Geometrically speaking, the absolute value is the distance between a number and the origin, even if the number is positive or negative.
It may seem convenient to say that the absolute value is obtained by removing any sign before the number. This may work in arithmetic, but can lead to errors in algebra. What is |x|, if the true value of x is unknown? x could be positive or negative. There is no visible sign to be removed. Another "safe" and purely algebraic definition of absolute value will be given later.
Absolute values are needed in the rules for operations of addition, subtraction, and multiplication of integers. It will be shown in this chapter that the integers form a system of numbers closed under subtraction, as well as addition and multiplication. These facts make the system of integers more useful than the systems of non-negative integers and natural numbers, which have more limitations.
[1.2] (Addition of signed integers)
(a) If two (or more) integers have the same sign, then add their absolute values and take the common sign for the sum.
(b) If two integers have different signs, then subtract their absolute values with the limited subtraction for natural numbers and take the sign of the integer with the larger absolute value.
Examples
(a) (+3) + (+2) = + (3 + 2) = 5; (-4) + (-5) = - (4 + 5) = 9.
(b) (+6) + (-4) = + (6 - 4) = 2; (-8) + (+5) = - (8 - 5) = 3.
Click here to see a physical example supporting the addition of signed integers.
[1.3] (Closure of the integers under addition) Any pair of integers may be added, and the sum is always an integer.
The the symbol for addition is the usual plus sign.. It is sometimes necessary to distinguish it from the plus sign used to indicate positive integers. Temporarily it will be written in red (+). For example (-6) + (+4) = -2. (For addition of natural numbers the addition sign should be also in red, but was not necessary until now in this section.)
The addition of integers shows the net effect of withdrawal and deposit of money to a bank account. If $4 are deposited and then $5 are withdrawn from the same account, the two operations together are equivalent to a withdrawal of $1 from that account.
Zero continues to play the role of an identity.
[1.4] (Zero is the additive identity for integers) For any integer n, n + 0 = n 0 + n = n.
If the sum of two integers is zero, then the two integers have a special relationship.
[1.5] (Additive inverses*) If the sum of two integers is zero then each is called the additive inverse of the other.
-4 is the additive inverse of 4, and 4 is the additive inverse of -4,
+5 is the additive inverse of -5 and -5 is the additive inverse of +5.
Counterpart numbers and additive inverses mean the same thing. Therefore, additive inverses always have the same absolute value, but have opposite signs.
The black minus sign is part of a negative integer: the minus sign (-) is part of -5. To indicate the additive inverse of an integer, a red minus sign will be written before the integer. Therefore, -(+3) = -3. -(-2) = +2. A red minus sign is different from a black minus sign that is part of a negative integer. Because the red minus sign is used to indicate additive inverses, the additive inverses are also called negatives of each other.
[1.6] The integers under addition are one of many algebraic systems. The four properties
(a) closure [1.3]
(b) associative operation [assumed]
(c) there is an identity element [1.4]
(d) every element has an inverse [1.5]
are found in many of those systems, so many, that they are worth collecting to define a category called a group.
The integers under addition have an additional property:
(e) commutativity assumed
Under addition the integers form a commutative group.
The even integers
The even natural numbers 2N are a system under addition not having properties (c) and (d), satisfying only half of (a),(b),(c),(d). Therefore 2N is called a semi-group. Actually 2N is a commutative semi-group.
The definition [1.13b] of subtraction of natural numbers can be extended literally to integers.
[1.7] (Definition of subtraction) The difference of one integer minus a second integer is that new integer such that the sum of the second integer and the new integer equals the first integer.
Notation: m - n = x if and only if n + x = m.
The difference solves the equation n + x = m where x is the unknown to be fouind. This definition is not very satisfory for computation. Instead the following can be used.
[1.8] Computing the difference) The difference of one integer minus a second integer can be computed by adding the additive inverse of the second integer to the first integer.
Notation: m - n = m + (additive inverse of n).
Examples:
(-3) - (+4) = (-3) + (-4)
(-6) - (-2) = (-6) + (+2)
Since [1.8] guarantees that subtraction for any pair of integers is possible, the following statement is a formal declaration of this fact.
[1.9] The system of integers is closed under subtraction.
The following is a very simple statement that will be used in the next section.
[1.10] Lemma The only integer that is equal to the sum of itself and itself is zero.
Notation: if n = n + n then n = 0.
the proof is as trivial as the lemma: simply add the additive inverse of n to both sides:
n + (-n) = n + n + (-n)
The left side becomes 0, the right side becomes n
0 = n.
[*]The product of zero and any negative integer is zero.
Let k be any positive integer. Then -k is a negative integer.
(0 + 0)(-k) may be evaluated two ways:
Since 0 + 0 = 0, then one way is 0(-k).
The distributive law produces 0(-k) + 0(-k)
Therefore equating these two results: 0(-k) = 0(-k) + 0(-k).
By the above lemma [1.10], 0(-k) = 0.
A similar argument shows that (-k)0 = 0.
This means that
[2.1] (Products with zero) if one (or both) of two integers is zero, then their product is zero.
Notation: For any integers m,n, if m = 0 or n = 0 then mn=0.
The converse of [2.1] is also true.
[2.2] (No divisors of zero) If the product of two integers is zero, then one or both must be zero.
Notation: For integers m,n, if mn = 0 then m = 0 or n = 0.
Click here to see a discussion supporting this statement.
This theorem supports the solving of quadratic equations like
0 = x2 - 5x + 6 = (x - 2)(x - 3)
Either of the factors of the product (x - 2)(x - 3) could be zero. It could be x - 2 = 0 or x - 3 = 0.
These give solutions x = 2 or x = 3.
Both of the above [2.1] and [2.2] may be combined and extended to products of any number of integers:
[2.3] A product of integers is zero if and only if at least one integer is zero (possibly more than one).
There remains a discussion of products involving negative integers. Again the various laws of numbers, especially the distributive law, are used as basis.
[#] If n and k are natural numbers, then n(-k) = - nk. [Product of a positive integer and negative integer is negative.]
The argument is simple and involves the evaluation of the expression
[##] If n and k are natural numbers, then (-n)(-k) = nk. [Product of two negative integers is positive.]
This time the expression to be evaluated two ways is
[2.4] (Product of two signed integers) The product of two integers is computed as follows:
(a) if one or both integers is zero, then the product is zero
the product of their absolute values is computed
(b) if both integers are positive, then the product is positive
(c) if an integer is positive and the other integer negative, then the product is negative
(d) if both integers are negative, then the product is positive
[2.5] (Products of two or more signed integers) The product of two or more integers is computed as follows:
<[2.5] (Integer powers of -1)
(-1)odd integer = -1
It will be shown that integers with negative integer exponents involve fractions, which are discussed in the next chapter.
(a) if any integer is zero, then the product is zero
the product of all their absolute values is computed
(b) if all integers are positive, then the product is positive
(c) if there are an odd number of negative integers, then the product is negative
(d) if there are an even number of negative integers, then the product is positive
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Positive exponents affect integers as they did non-negative integers.
(-3)2 = (-3)(-3) = +9
(-3)
It is obvious that (-3) raised to an even power will always be positive. This is true for any negative integer raised to an even power.
(-2)3 = (-2)(-2)(-2) = -8
(-2)5 = (-2)(-2)(-2)(-2)(-2) = -32
It is obvious that (-2) raised to an odd power will always be negative. This is true for any negative integer raised to an odd power.
The signs can be summerised as follows:
(-1)even integer = +1