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EXPLANATION OF THE ALGEBRAIC METHOD OF NÓTATION :DEFINITIONS AND AXIOMS.

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1. Algebra is a general method of computation, in which abstract quantities and their several relations are made the subject of calculation, by means of alphabetical letters and other signs.

2. The letters of the alphabet may be employed at pleasure for denoting any quantities, as algebraical symbols or abbreviations; but, in general, quantities whose values are known or determimed, are expressed by the first letters, a, b, c, &c.; and unknown or undetermined quantities are denoted by the last or final ones, u, v, w, x, &c.

3. Quantities are equal when they are of the same magnitude. The abbreviation a=b implies that the quantity denoted by a is equal to the quantity denoted by b, and is read a equal to b; ab or a greater than b, that the quantity a is greater than the quantity b: and ab or a less than b, that the quantity a is less than the quantity b.

4. Addition is the joining of magnitudes into one sum. The sign of addition is an erect cross; thus, a+b implies the sum of a and b, and is called a plus b. if a represent 8 and b; 4; then, ab represents 12, or a+b=8+4=12.

5. Subtraction is, the taking as much from one quantity as is equal to another. Subtraction is denoted by a single line; as a-b or a minus b, which is the part of a remaining, when a part equal to 6 has been taken from it; if a=9, and b=5; a-b expresses 9 diminished by 5, which is equal to 4, or Eb6=9-5=4.

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6. Also, the difference of two quantities a and b ; when it is not known which of them is the greater, is represented by the sign ; thus, ab is a-b, or b -a; and ab signifies the sum or difference of a and b.

7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple apposition; thus, a Xb, a. b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted by b; if a=5 and b=7; then aXb=5×7=35, or a. b=5.7=35, or ab=5X7=35.

Scholium. The multiplication of numbers cannot be expressed by simple apposition. A unit is a magnitude considered as a whole complete within itself. And a whole number is composed of units by continued additions; thus, one plus one composes two, 2+1=3,3+1=4, &c.

8. Division is the subtraction of one quantity from another as often as it is contained in it; or the finding of that quotient, which, when multiplied by a given divisor, produces a given dividend.

Division is denoted by placing the dividend before the sign , and the divisor after it; thus ab, implies that the quantity a is to be divided by the quantity b. Also, it is frequently denoted by placing one of the two quantities over the other, in the form of a fraction; thus, a÷b; if a=12, b=4 ;

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9. A simple fraction is a number which by continual addition composes a unit, and the number of such fractions contained in a unit, is denoted by the denominator, or the number below the line; thus, +}+}=1. A number composed of such simple fractions, by continual addition, may properly he termed a multiple fraction; the number of simple fractions composing it, is denoted by the upper figure or In this sense,,,, are multiple fractions; and 3=1, }=3+}=1+1=1}.

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10. When any quantities are enclosed in a parenthesis, or or have a line drawn over them, they are considered as one quantity with respect to other symbols; thus a-(b+c), or a-b+c; implies the excess of a above the sum of b and c ; Let a 9, 6-3, and c=2; then, a-(b+c)=9-(3+2)=9

Also, (a+b)x

—5—4, or a−b+c=9—3+2=9—5—4. (c+d), or a+b Xc+d,denotes that the sum of a and b is to be multiplied by the sum of c and d ; thus, let a=4, b=2. c=3, and d=5; then (a+b)x(c+d)=(4+2) × (3+5)=6×8=48, or a+bXc+d=4+2×3+5=6X8=48. And (a - b)÷(c +d,) or implies the excess of a above b, is to be divided by the sum of c and d ; if a=12, b=2, c=4, and d=1 ; then, (a—b)÷(c+d)=(12—2)÷(4+1)=10÷5=2,or c+d

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The line drawn over the quantities is sometimes called a vinculum.

11. Factors are the numbers or quantities, from the multiplication of which, the proposed numbers or quantities are produced; thus, the factors of 35 are 7 and 5, because 7X5 =35; also, a and b, are the factors of ab ; 3, a2, b and c2, are the factors of 3a2bc2; and a+b and a-b are the factors of the product (a+b) x (a—b).

When a number or quantity is produced by the multiplication of two or more factors; it is called a composite number or quantity; thus, 35 is a composite number, being produced by the product of 7 and 5; also, 5acx is a composite quantity, the factors of which are 5, a, c, and x.

12, When the factors are all equal to each other, the product is called a power of one of the factors, and the factor is called the root of the product or the power. When there

are two equal factors, the product is called the second power or square of either factor, and the factor is called the second root or square root of the power. When there are three equal factors, the product is called the third power or cube of either factor, and the factor is called the third root or cube root of the power. And so on for any number of equal factors.

13. Instead of setting down in the manner of other products, the equal factors which multiplied together constitute a power, it is evidently more convenient to set down only one of the equal factors, (or, in other words, the root of the power,) and to designate their number by small figures or letters placed near the root. These figures or letters are always placed at the upper and right side of the root, and are called the indices or exponents of the power.

For example:

axa xa xa or aaaa is denoted thus, a^;
yxy XyXyxy or yyyyy, thus, y5;

6. Also, the difference of two quantities a and b ; when i is not known which of them is the greater, is represented by the sign; thus, ab is a-b, or b-a; and ab signifies the sum or difference of a and b.

7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple apposition; thus, a Xb, a . b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted by b; it a=5 and b=7; then a×6=5×7=35, or a,b=5.7=35, or ab=5X7=35.

Scholium. The multiplication of numbers cannot be expressed by simple apposition. A unit is a magnitude considered as a whole complete within itself. And a whole number is composed of units by continued additions; thus, one plus one composes two, 2+1=3,3+1=4, &c.

8. Division is the subtraction of one quantity from another as often as it is contained in it; or the finding of that quotient, which, when multiplied by a given divisor, produces a given dividend.

Division is denoted by placing the dividend before the sign , and the divisor after it; thus ab, implies that the quantity a is to be divided by the quantity b. Also, it is frequently denoted by placing one of the two quantities over the other, in the form of a fraction; thus, &=a÷b; if a=12, b=4

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9. A simple fraction is a number which by continual addition composes a unit, and the number of such fractions contained in a unit, is denoted by the denominator, or the num ber below the line; thus, +{+}=1. A number com posed of such simple fractions, by continual addition, may properly be termed a multiple fraction; the number of simple fractions composing it, is denoted by the upper figure or numerator. In this sense,,,, are multiple fractions; and 1,3+}=1+1=1}.

10. When any quantities are enclosed in a parenthesis, or or have a line drawn over them, they are considered as one quantity with respect to other symbols; thus a-(b+c), or a-b+c; implies the excess of a above the sum of b and c ; Let a 9, 6-3, and c=2; then, a-(b+c)=9-(3+2)=9

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