ARITHMETIC.

Arithmetic is the science that treats of numbers, and of the methods of computing by means of them.

"Notation" is a method of writing numbers by characters or figures. The number ten is the basis of Our system of notation, containing ten numeral figures or "digits," 1, 2, 3, 4, 5, 6. 7, 8, 9, 0, the last, the cipher or zero, having no value except in combination. This system is known as the Arabic. (The Roman system uses the capital letters, I for 1, V for 5, X for 10, L for 50, C for 100, D-for 500, M. for 1,000. An equal or a smaller figure placed after a bigger one is added thereto; a smaller one placed in front is subtracted therefrom. Thus, 1888 is represented as follows: MDCCCLXXXVIII; 67 = LXVII, 43= XLIII. The use of this system is limited.)

"Numeration" is the art of reading figures employed to express numbers. The following table shows the places of the figures, which are grouped in periods of three figures each, counting from the right, commonly separated by commas.

This is read two hundred and twenty-five billion nine hundred and ten million six hundred and seventy-three thousand four hundred and eighty-five.

Tens

Units

Hundreds

∞ Tens

& Units

Addition is the method of finding the "sum" of two or mor given numbers. The sign of addition is +, reads "plus," and signifies "more."

"Equation" is an expression of equality of two numbers. The sign of equation is =, it reads “equals” or “equal to." Thus 3 + 4 = 7, reads 3 plus 4 equals 7.

Subtraction is the method of finding the "difference" between two given numbers.

"Minuend" is the greater of the two numbers.

"Subtrahend" is the smaller of the two numbers.

"Difference" or "remainder" is the result obtained by subtracting. The sign of subtraction is, reads "minus" and signifies "less." Thus 13 - 8 is read "13 minus 8," and signifies that 8 is to be subtracted from 13.

Multiplication is a method of finding the result produced by a given number taken a given number of times.

"Multiplicand" is the number to be multiplied.
"Multiplier" is the number by which to multiply.

"Product" or "multiple" is the result of multiplication. Multiplicand and multiplier are called the factors of the product. The sign of multiplication is X, reads "times" or "multiplied by." Division is the method of finding how often one given number contains another.

"Dividend" is the number to be divided.

"Divisor" is the number by which to divide.

"Quotient" is the result of the division. The sign of division is, reads "divided by." Division is also indicated by placing the dividend above the divisor, with a line between them. Thus sa is read "63 divided by 7," in which 63 is the dividend, and 7 the divisor.

63

Properties of Numbers. An "integral number" or "integer" is a number representing whole things. It is either even or odd, prime or composite. "Even numbers" are divisible by 2: "odd numbers" are not exactly divisible by 2. 2, 6, 12, 14, etc., are even numbers. 3. 7. 13, 15, etc., are odd numbers.

"Prime Number" is a number which has no integral factors except unity and itself. 2, 3, 5, 7, 11, etc., are prime numbers. "Composite Number" is a number that has other integral fac

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tors besides unity and itself. Thus 24 is a composite number, ne since 24-8X 3.

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"Factors" of a number are the numbers which multiplied he together will produce such number. 9 and 7 are factors of 63. "Prime Factor" is a prime number used as a factor, and is also the prime divisor of it; thus 3 and 5 are prime factors of 15, and en prime divisors.

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nd

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"Exact Divisor" of a number is one that will divide that number without a remainder. 7 is an exact divisor of 63. Exact divisors of a number are also the factors of that number.

Numbers are "prime to each other" when they have no common integral factor or divisor. 7 and 16 are prime to each other. "Factoring" is the resolving of a composite number into its factors, and is done by division.

To find the prime factors of a composite number: Divide the by given number by any prime factor of it, and the resulting quotient by another, and continue the division until the quotient is a prime number. The several divisors and the last quotient are the prime factors.

2, 3, 5, 7, II

II

The product of all the prime factors is the given number. A "Common Divisor" of two or more numbers is a divisor of each of them, and also a common factor of each of them. The "Greatest Common Divisor" of two or more numbers is the greatest "common factor," and is the product of all the common prime factors.

To find the greatest common divisor of two or more numbers: Resolve the given numbers into their prime factors; select the factors which are common, and multiply them together. The product will be the greatest common divisor. The greatest common divisor of 42 and 112 is:

"Multiple" of a number is a number exactly divisible by the given number. 6 is a multiple of 2 and 3. A “common multiple" of two or more given numbers is a number exactly divisible by each of them. The "least common multiple" is the least number exactly divisible by each of them.

To find the least common multiple of two or more numbers: Resolve the given numbers into their prime factors; select all the different factors, taking each the greatest number of times it is found in any of the numbers, and multiply together the factors thus selected:

The least common multiple of 10, 45, 75, 90 is:

10 = 2 X 5

453 X3 X5

75 3 X5 X 5

902X 3X3 X5

and 2 X 3X3 X5 X5= 450

Another method is to write the numbers in an horizontal line, omitting such of the smaller numbers as are factors of the larger, and draw a vertical line at the left. Divide by any prime factor that will exactly divide two or more of the given numbers, and write the quotients and undivided numbers in a line underneath. Divide the quotients and undivided numbers until they are prime to each other. The product of the divisors and the final quotients and undivided numbers is the least common multiple.

"Cancellation" is the process of abridging operations in division by rejecting equal factors from both dividend and divisor. Divide 13 X7X5X3 by 3 X5 X 7. Then

FRACTIONS.

If unity be divided into any number of equal parts, one or more of these parts is called a "fraction."

There are two kinds of fractions: "Common" or "vulgar" fractions, commonly called "fractions" simply, and "decimal" fractions, commonly called "decimals."

A common fraction is represented by two numbers, called "terms," which are written one above, the other below an horizontal or slanting line, thus: 2, 3, 4, 5%.

"Denominator" of a fraction is the number of equal parts into which the unit is divided, and is written below the line. Thus in the denominator is 4, showing that the unit is divided

into 4 equal parts.

"Numerator" of a fraction is the number of equal parts taken to form the fraction, and is written above the line. Thus: in the numerator is 5, showing that 5 of the 6 equal parts are

taken or expressed by the fraction.

"Proper fraction" is one whose numerator is less than its denominator; as 3, 4, 7%.

"Improper fraction" is one whose numerator is equal to or greater than the denominator; as 1. 7, 9.

"Mixed number" is an integer and fraction united; as 44, 15%.

REDUCTION OF FRACTIONS.

By "reduction" the form is changed, the value remaining the same. Fractions are changed to higher terms by multiplication, to lower terms by division.

Reduction to lower terms: 12 = 1 = 3
Reduction to lowest terms: 1 = 34

where numerator and denominator are prime to each other.

To Reduce an Integer or a Mixed Number to an Improper Fraction: Multiply the integer by the required denominator, and to the product add the numerator of the fraction, and under the result write the required denominator.

To Reduce an Improper Fraction to an Integer, or a Mixed Number. Divide the numerator by the denominator, 3=4=4'2