Architecture

# Any Non Negative Numeral Less Than Ten, Cornerstone

l>Non-Negative Integers B2Go to other chapters or to other volumes of mathAdditional Material for this ChapterExercises for this ChapterAnswers to ExercisesComputer Programs for this ChapterReturn to index of all chapters in this volume Volume B Chapter 2Non-negative IntegersStatements with an asterisk (*) will be generalized in later sections to apply to larger systems of numbers. The associative, commutative and distributive laws are assumed to be true for the system discussed in this section. It is an interesting fact that the system of Roman numerals did not contain a symbol for zero. But zero will play significant roles in many discussions below and in future volumes. It is necessary to have a symbol to indicate nothing, 0, called zero. For example, an empty bag has 0 objects in it. Zero is a new number to be “attached” to the set of natural numbers. The set N0 of non-negative integers is the union of zero and the set N of natural numbers. N0 = {0} ∪ N = {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,… } The position of zero in set N0 indicates that zero will be less than any natural number, and is the smallest non-negative integer.Since the set N is a subset of the set N0 (N ⊂ N0), many of the laws and operations on natural numbers may be extended to non-negative integers.

### Section 1: Addition of non-negative integers

It is easy to extend the addition operation on N to an addition operation on N0. The addition of natural numbers was discussed in the previous chapter. There remains only to determine addition involving zero. The following does this:<1.1> (Additive identity*) Any sum of a non-negative integer and zero remains the same if zero is removed from the sum.

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### Section 2: Multiplication of non-negative integers

The next task is to extend the operation of multiplication to the non-negative integers. Many of the statements about multiplication of natural numbers were presented as duals of statements about addition of natural numbers. Therefore, many of the following statements are duals of statements about addition of non-negative integers. Therefore, it is desirable that the commutative, associative and distributive laws be true for non-negative integers. The following is the dual of <1.1>:<2.1> (Multiplicative identity*) For any non-negative integer n,1⋅n = n and n⋅1 = n.<2.2> (Multiplicative identity is unique) There is only one multiplicative identity, 1.Notation: if nx=n for all nonnegative integers n, then x=1.Suppose 1″ is also a multiplicative identity for the non-negative integers. Then the product (1)(1″) can be evealuated two ways:(1)(1″) = 1″ (1)(1″) = 1 Therefore 1″ = 1 because both are equal to the same expression (1)(1″).In order to extend some laws about multiplication of natural numbers to non-negative integers, it is necessary sometimes to focus attention on the behavior of zero when multiplying non-negative integers. There is no duality here.It is already known that the product of a natural number and zero is equal to zero: n0 = 0. Algebra and geometry provide motivation for this equation. Simply add n zeros together: 0 + 0 + … + 0 = 0.

Consider the rectangles in the adjacent figures where AB = n. In Fig 1 the area of the rectangle is length x height = AB x BC = n x BC. Now let vertical sides BC and AD shrink to zero, so that roof DC collapses onto floor AB in Fig 2. Hence BC=0. Then 0 = area = AB x 0 = n0.The following and its proof extends the idea to non-negative integers:<2.3> (Multiplication by zero*) Any product involving zero and non-negative integers is zero. Notation: For any non-negative integer n, n0 = 0 and 0n = 0. In order that the two-sided distributative law hold for non-negative integers, it is necessary that(0 + 0)n = 0n + 0n, n(0 + 0) = n0 + n0 for any non-negative integer nBut 0 + 0 = 0. So these two equations reduce to0n = 0n + 0n, n0 = n0 + n0Subtracting 0n and n0 from these quations respectively produces0 = 0n, 0 = n0Actually <1.7> is sufficient to ensure that the two-sided distributive law holds for non-negative integers if it already holds for natural numbers.<2.4> (Closure under multiplication) The product of any non-negative integers is a non-negative integer.Notation: if m,n are non-negative integers, then their product mn is a non-negative integer.Clearly, <2.4> is the dual of <1.2>. Some of the following about multiplication are duals of above statements about addition.The fact that multiplication of natural numbers is commutative and <2.3> together imply:<2.5> (Multiplication is commutative) The product of two non-negative integers is the same no matter what the order of the numbers is.Notation: For any non-negative integers m,n, mn = nm.The proof consists of two parts.Part I: m,n are both natural numbers. The commuttive laws for natural numbers were discussed in the previous chapter 1.Part II: m=0 or n=0. mn = 0 = nm (by 2.3).The fact that multiplication of natural numbers is associative and <2.3> together imply:<2.6> (Multiplication is associative) For the product of three non-negative integers, either multiplication may be done first.Notation: For any non-negative integers k,m,n, k(mn) = (km)n.The proof consists of four parts. In each part <2.3> is used repeatedly.Part I: k,m,n are all natural numbers. The associative for natural numbers was discussed in the previous chapter 1.Part II: k=0. k(mn) = 0(mn) = 0 = 0m = (0m)n = (km)n.Part III: m=0. k(mn) = k(0n) = k0 = 0 = 0n = (k0)n = (km)n.Part IV: n=0. k(mn) = k(m0) = k0 = 0 = (km)0 = (km)n.The associative law allows the definition of the multiplication of three non-negative integers without parentheses: kmn. The statements <2.5> and <2.6> combine to support <2.7> (Multiplication in general) The multiplication of any non-negative numbers may be done in any orderNotation (for three numbers): kmn = knm = mkn = mnk = nkm = nmk.For h number of non-negative integers, there are h! (factorial) products of the numbers in different orders. (Factorials were discussed briefly in Chapter on permutations in Volume B.)

### Section 3: Additional properties of non-negative integers

——Given two non-negative integers it is possible to produce two more non-negative integers from them using the two operations of addition and multiplication of the two numbers: m + n and and mn. The division algorithm, to be discussed here, provides another way to produce the two non-negative integers. It is based on an attempt to divide the second number into the first, using only non-negative integers. But division by zero is not allowed. So the second number must be a natural number. It may be helpful to write the division as a ratio:(non-negative integer)/(natural number) even though a discussion of fractions is delayed until Chapter 4. For that reason here it is called a ratio. Then the non-negative integer becomes the numerator and the natural number becomes the denominator of the ratio. (Some people prefer to use the notation(non-negative integer) : (natural number)as the symbol for ratio.)Suppose the two given numbers are 13 and 3. The ratio is the symbol 13/3. Long division has the form

Notice that it would be impossible to do the division if the natural number 3 were replaced by zero.In the following, may the reader think of the given numbers (non-negative integer) and (natural number) as forming a ratio (non-negative integer) / (natural number). The (quotient) and (remainder) are computed by the method of long division shown above. The following is a very important theorem involving non-negative integers. Intuitively speaking, it is an attempt to do division using only non-negative integers and no decimals nor fractions.<3.4>(Division algorithm) For any (non-negative integer) and any (natural number) , there exist unique non-negative integers (quotient) and (remainder) all satisfying non-negative integer = (natural number)x(quotient) + remainder The remainder is always less than the (natural number).Notation: if m is any non-zero integer, and if n is any natural number, then there exist unique non-negative integers q and r such thatm = n(q) + r where 0 r The inequalities 0 r r is redundant, because r must be a non-negative integer.Click here to see arguments supporting <3.4>.Examples: ratio m/n For m and n, there exist unique numbers q and r that satisfy m = n(q) + r and 0rn. For 13 and 3, there exist unique numbers 4 and 1 that satisfy 13 = 3(4) + 1 and 013. For 19 and 5, there exist unique numbers 3 and 4 that satisfy 19 = 5(3) + 4 and 045. For 3 and 5, there exist unique numbers 0 and 3 that satisfy 3 = 5(0) + 3 and 035. For 8 and 4, there exist unique numbers 2 and 0 that satisfy 8 = 4(2) + 0 and 004. For 7 and 0, there do not exist any non-negative integers q and r that satisfy 7 = 0(q) + r and 0r0. For more numerical examples of the division algorithm click here.For the ratio (non-negative integer)/(natural number): (a) The natural number is a divisor of the non-negative integer if and only if the remainder = 0. (b) If the natural number is larger than the non-negative integer then quotient = 0 and remainder = non-negative integer.

A geometric interpretation may increase understandinggiven non-negative integer = 39 and natural number = 9. 39 = 9(4) + 3.If each number in the set M = N0 of all non-negative integers is multiplied by 9, then an infinite subset M9 of all (non-negative) multiples of 9 is obtained:M9 = {0,9,18,27,36, 45,54,63,72,81,90,99,…}39 is between 36 and 45 in the set M9 of multiples of 9. The largest multiple less than 39 is 36. And 39 is 3 beyond that largest multiple 36. This can be seen in the adjacent figure.