Artificial numbers

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/4 is as natural as 2, and sq.root of 5 as naturally comes into use as 5, although we cannot look at an object 3/4 of a time nor can we pick up a book sq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Further, on the algebraic scale we can represent such fractions as 1/2 and -3/4, and such surd numbers as -sq. root of 2 and 3 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/b, in which a and b are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2 and 3 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1 are called complex numbers.

Uses of Irrational Numbers.

In the solution of quadratic equations, excepting those artificially constructed to give only integral values of the unknown quantity, the roots are generally irrational, not being expressible as the quotient of a/b, where a and b are integers. This is seen in the simple quadratic x²=2, where . Since early arithmetic was concerned largely with rational numbers, the irrational ones were generally assigned to algebra, the branch of mathematics in which they were needed. The purpose of placing them there was soon obscured, however, the result being a much more extensive treatment of the subject than was warranted by any practical considerations. If, in a physical problem, there is need for solving the equation x²=243, it is important to find the value of x to a definite degree of approximation. If it is stated that , nothing is gained by writing this result as . An expression like and , but (for no very good reason), not like , is known as a surd, from a medieval Latin translation of an Arabic rendering of the Greek al'ogos (irrational). Until recently the finding of the square root of a binomial surd, as of or of , was a familiar operation in elementary algebra, and there is still good cause for complaint that the work in surds is excessive. With the properties of such transcendental numbers as e and , elementary algebra is little concerned.

Factors.

The factorizing of polynomials has a place in the theory of equations and in the advanced study of polynomials, but its value in elementary algebra is slight. In the algebra that the pupil will use in the sciences or in mechanical arts its legitimate place is not large. The need of the pupil are usually met by the cases of monomial factors, and of the binomial factors of the expressions of the type x²+(a+b)x+ab and x²-y².

The original idea, carried over from arithmetic, was that factors should be integral and rational. In practical use, however, it is often necessary to enlarge this conception, and to speak of x²-I/a² as having the factors x+I/a and x-I/a, of and as factors of ; and of as factors of .

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/43/4 is as natural as 22, and sq.root of 5sq.root of 5 as naturally comes into use as 5 5, although we cannot look at an object 3/43/4 of a time nor can we pick up a book sq.root of 5 timessq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3-3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/42/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Algebraic Scale.

Algebraic Scale.

If we construct a scale of integers, 1, 2, 3, 4, ...,1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0.... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 00 on the scale, to 1, 2, 3, 4, ... 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ...1, 2, 3, 4, ... below zero nn a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., 1, 2, ..., we may write them as +1, +2, ..., +1, +2, ..., although even without any sign they are considered positive.

Further, on the algebraic scale we can represent such fractions as 1/2 1/2 and -3/4-3/4, and such surd numbers as -sq. root of 2 -sq. root of 2 and 3 sq. roots of 53 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/ba/b, in which aa and bb are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2-sq. root of 2 and 3 sq. roots of 53 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1a+b sq. root of -1 are called complex numbers.

Algebraic Expressions.

Algebraic Expressions.

An expression consisting of a single letter, or made up either of letters or of letters and numerals, combined so as to represent some or all of the operations of addition, subtraction, multiplication, division, involution (the finding of powers) and evolutions (the finding of roots), is an algebraic expression. If it does not involve addition or subtraction, it is a monomial; but in the expression a-(b+c), -(b+c)a-(b+c), -(b+c) is considered as a monomial, and so in other similar cases involving signs of aggregation (parenthesis, brackets, etc.). If an algebraic expression is not a monomial, it is a polynomial, the binomial (two-term) and trinomial (three-term)being special types. In algebra, the letters of an expression represent numbers of some kind. In the monomial ab, a and b are factors of expression. If the value of either factor, say aa, is known and is to remain the same throughout the discussion of an expression, it is called a constant; but if it may have any value we please to give it and change from one value to another, it is called a variable. Constants are often represented by the first letters of the alphabet (a, b, c, ...)(a, b, c, ...) and variables by the last letters (..., x, y, z) (..., x, y, z), but this is not a universal rule especially in physical formulas. In an equation, say 2a-x=4, x2a-x=4, x usually represents a number to be found - "the unknown quantity" - the first letters of the alphabet representing numbers supposed to be known. In the monomial 2ax**22ax², we may speak of any factor as the coefficient of the rest of expression, but it is customary to speak of 2 2 as a numerical coefficient, and of 2a2a as the coefficient of x**2x², the coefficient being the first factor or factors. For example, in the expression 1/2(a+b)1/2(a+b)x**2y**3x²y³, 1/2, 1/2 is the coefficient of (a+b)(a+b)x**2y**3x²y³, and 1/2(a+b)1/2(a+b) is the coefficient of x**2y**3x²y³, and it is allowable to speak of 1/2(a+b)1/2(a+b)x**2x² as the coefficient of y**3y³ and so on. In the expression x**3x³, 3is3 is the exponent of xx, and similarly in the case of x**m.x^m. In the expression mxmx, the coefficient mm (if it be an integer) represents the number of times that xx is taken as an addend; while in the expression x**mx^m, m, m (if it be an integer) represents the number of times that xx is taken as a factor. In each case the meaning is later extended to permit of mm being any kind of number (fractional, surd, imaginary, etc.)

Function.

An expression like 2x+5 is called a function of x and is said to "depend upon" x for its value. Similarly, 2x+y³ is a function of x and y. For brevity we may, in any discussion, write f(x) for the function of x, and f(x,y) for the function of x and y, and so on for other variables. We may then, in discussing x-3, for example, say that f(i)=i-3=-2, f(7)=7-3=4, f(-2)=-2-3=-5, f(0)=0-3=-3, and so on, according to the value of x which we substitute in x-3.

The Elementary Operations.

From the standpoint of actual use, whether in the natural sciences or in pure mathematics, there is little need for the ordinary operations involving polynomials, therefore a brief treatment of the four fundamental operations, limited chiefly to binomial operators, with a slight reference to the theory of roots, is all that is essential to the further study of the science of algebra. The operations upon algebraic fractions were adapted from arithmetic after the introduction of the improved symbolism of the 17th century. They later became more complicated owing to the doctrine of formal discipline, the result leading to an expenditure of time quite out of proportion to the use made of them. In the 19th century the time consumed in reducing artificial fractions to lower terms, and in operations involving polynomials was excessive.

Ratio.

In practical work a ratio is considered simply as a fraction, although fundamentally the ratio of a circumference to the diameter of a circle is a transcendental number and not a fraction, as we define the term. Practical work in a laboratory or workshop is not considered with irrationals as such; it seeks for precision within certain designated limits, recognizing that all measurement is approximate. On this account and because of the immaturity of the pupils, elementary algebra looks upon a proportion as a fractional equation, and deals with all rations as simple functions, ignoring the distinction between algebraic and arithmetical fractions, and the fact that ratio may be irrational.

Algebraic Scale.

Algebraic Scale. If we construct a scale of integers, 1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 0 on the scale, to 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ... below zero n a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., we may write them as +1, +2, ..., although even without any sign they are considered positive.

Algebraic Expressions.

An expression consisting of a single letter, or made up either of letters or of letters and numerals, combined so as to represent some or all of the operations of addition, subtraction, multiplication, division, involution (the finding of powers) and evolutions (the finding of roots), is an algebraic expression. If it does not involve addition or subtraction, it is a monomial; but in the expression a-(b+c), -(b+c) is considered as a monomial, and so in other similar cases involving signs of aggregation (parenthesis, brackets, etc.). If an algebraic expression is not a monomial, it is a polynomial, the binomial (two-term) and trinomial (three-term)being special types. In algebra, the letters of an expression represent numbers of some kind. In the monomial ab, a and b are factors of expression. If the value of either factor, say a, is known and is to remain the same throughout the discussion of an expression, it is called a constant; but if it may have any value we please to give it and change from one value to another, it is called a variable. Constants are often represented by the first letters of the alphabet (a, b, c, ...) and variables by the last letters (..., x, y, z), but this is not a universal rule especially in physical formulas. In an equation, say 2a-x=4, x usually represents a number to be found - "the unknown quantity" - the first letters of the alphabet representing numbers supposed to be known. In the monomial 2ax**2, we may speak of any factor as the coefficient of the rest of expression, but it is customary to speak of 2 as a numerical coefficient, and of 2a as the coefficient of x**2, the coefficient being the first factor or factors. For example, in the expression 1/2(a+b)x**2y**3, 1/2 is the coefficient of (a+b)x**2y**3, and 1/2(a+b) is the coefficient of x**2y**3, and it is allowable to speak of 1/2(a+b)x**2 as the coefficient of y**3 and so on. In the expression x**3, 3is the exponent of x, and similarly in the case of x**m. In the expression mx, the coefficient m (if it be an integer) represents the number of times that x is taken as an addend; while in the expression x**m, m (if it be an integer) represents the number of times that x is taken as a factor. In each case the meaning is later extended to permit of m being any kind of number (fractional, surd, imaginary, etc.)

The numbers used in counting objects have long been known as natural numbers. Numbers that are not naturally used in counting objects are commonly called artificial numbers, a term that is open to certain objections. From various points of view the number 3/4 is as natural as 2, and sq.root of 5 as naturally comes into use as 5, although we cannot look at an object 3/4 of a time nor can we pick up a book sq.root of 5 times. The term is convenient, however, even though it may be as inappropriate as "imaginary" in connection with sq. root of -3, or "fraction" (in its primitive sense) in connection with 2/2, 5/2, 1/2/3/4. The preceding discussion leads to a much larger and more important question as to the meaning of number itself.

Algebraic Scale. If we construct a scale of integers, 1, 2, 3, 4, ..., and count backwards (to the left) by repeatedly subtracting 1, we have ... 4, 3, 2, 1, 0. The next count on the scale carries us beyond the point marked 0 on the scale, to 1, 2, 3, 4, ... in the opposite direction, just as we count 1, 2, 3, 4, ... below zero n a thermometer. For obvious reasons we find it convenient to speak of these symbols on the other side of zero as representing numbers, even though we cannot look at an object "three-less-than-zero" or "three-on-the-other-side-of-zero" times. We were led to these particular artificial numbers by subtracting as we went down the scale, and so we come to designate them a minus sign, thus giving to that sign a qualitative meaning (say "negativeness") instead of the operative one (subtraction). Such numbers are imaginary in some senses and natural in others. If we wish to emphasize the positive nature of 1, 2, ..., we may write them as +1, +2, ..., although even without any sign they are considered positive.

Further, on the algebraic scale we can represent such fractions as 1/2 and -3/4, and such surd numbers as -sq. root of 2 and 3 sq. roots of 5, so these are as real in certain situations as integers themselves.

In the expression a/b, in which a and b are natural numbers, does not denote a natural number, it is called a fraction, but since division by zero has no meaning, the case in which b = 0 is excluded. Integers and fractions are classified as rational numbers. More generally speaking, a rational algebraic fraction is the quotient of any integral function by any other integral function. Certain numbers such as -sq. root of 2 and 3 sq. roots of 5 do not come within a definition of rational numbers, and are called irrational numbers. In elementary algebra we also meet with certain numbers represented by the symbol a sq.root of -1, and these are called imaginary numbers. Numbers like a+b sq. root of -1 are called complex numbers.