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SUPPLEMENT

DEFINITIONS, PRINCIPLES, AND RULES

A Unit is a single thing.
A Number is a unit or a collection of units.
The Unit of a Number is one of that number.
Like Numbers are those that express units of the same kind.
Unlike Numbers are those that express units of different kinds.
A Concrete Number is one in which the unit is named.
An Abstract Number is one in which the unit is not named.
Notation is expressing numbers by characters.
Arabic Notation is expressing numbers by figures.
Roman Notation is expressing numbers by letters.
Numeration is reading numbers expressed by characters.
The Place of a Figure is its position in a number.

A figure standing alone, or in the first place at the right of other figures, expresses ones, or units of the first order.

A figure in the second place expresses tens, or units of the second order.

A figure in the third place expresses hundreds, or units of the third order ; and so on.

A Period is a group of three orders of units, counting from right to left.

RULE FOR NOTATION. — Begin at the left, and write the hundreds, tens, and units of each period in succession, filling vacant places and periods with ciphers.

RULE FOR NUMERATION. Beginning at the right, separate the number into periods.

Beginning at the left, read the numbers in each period, giving the name of each period except the last.

ADDITION

Addition is finding a number equal to two or more given numbers.

Addends are the numbers added.
The Sum, or Amount, is the number obtained by addition.

PRINCIPLE. — Only like numbers, and units of the same order can be added.

RULE. — Write the numbers so that units of the same order shall be in the same column.

Beginning at the right, add each column separately, and write the sum, if less than ten, under the column added.

When the sum of any column exceeds nine, write the units only, and add the ten or tens to the next column.

Write the entire sum of the last column.

SUBTRACTION

Subtraction is finding the difference between two numbers.
The Subtrahend is the number subtracted.
The Minuend is the number from which the subtrahend is taken.

The Remainder, or Difference, is the number left after subtracting one number from another.

PRINCIPLES. Only like numbers and units of the same order can be subtracted.

The sum of the difference and the subtrahend must equal the minuend.

RULES. — I. Write the subtrahend under the minuend, placing units of the same order in the same column.

Beginning at the right, find the number that must be added to the first figure of the subtrahend to produce the figure in the corresponding order of the minuend, and write it below. Proceed in this way until the difference is found.

If any figure in the subtrahend is greater than the corresponding figure in the minuend, find the number that must be added to the former to produce the latter increased by ten; then add one to the next order of the subtrahend and proceed as before.

II. Beginning at the units' column, subtract each figure of the subtrahend from the corresponding figure of the minuend and write the remainder below.

If any figure of the subtrahend is greater than the corresponding figure in the minuend, add ten to the latter and subtract; then, (a) add one to the next order of the subtrahend and proceed as before; or, (b) subtract one from the next order of the minuend and proceed as before.

MULTIPLICATION

Multiplication is taking one number as many times as there are units in another number.

The Multiplicand is the number taken or multiplied.

The Multiplier is the number that shows how many times the multiplicand is taken.

The Product is the result obtained by multiplication.

PRINCIPLES. The multiplier must be an abstract number. The multiplicand and the product are like numbers.

The product is the same in whatever order the numbers are multiplied.

RULE. — Write the multiplier under the multiplicand, placing units of the same order in the same column.

Beginning at the right, multiply the multiplicand by the number of units in each order of the multiplier in succession. Write the figure of the lowest order in each partial product under the figure of the multiplier that produces it. Add the partial products.

To multiply by 10, 100, 1000, etc.

RULE. — Annex as many ciphers to the multiplicand as there are ciphers in the multiplier.

DIVISION

Division is finding how many times one number is contained in another, or finding one of the equal parts of a number.

The Dividend is the number divided.
The Divisor is the number contained in the dividend.
The Quotient is the result obtained by division.

PRINCIPLES. — When the divisor and the dividend are like numbers, the quotient is an abstract number.

When the divisor is an abstract number, the dividend and the quotient are like numbers.

The product of the divisor and the quotient, plus the remainder, if any, is equal to the dividend.

RULE. — Write the divisor at the left of the dividend with a line between them.

Find how many times the divisor is contained in the fewest figures on the left of the dividend, and write the result over the last figure of the partial dividend. Multiply the divisor by this quotient figure, and write the product under the figures divided. Subtract the product from the partial dividend used, and to the remainder annex the next figure of the dividend for a new dividend.

Divide as before until all the figures of the dividend have been used.

If any partial dividend will not contain the divisor, write a cipher in the quotient, and annex the next figure of the dividend.

If there is a remainder after the last division, write it after the quotient with the divisor underneath.

FACTORING An Exact Divisor of a number is a number that will divide it without a remainder.

An Odd Number is one that cannot be exactly divided by two. An Even Number is one that can be exactly divided by two.

The Factors of a number are the numbers that multiplied together produce that number.

A Prime Number is a number that has no factors.
A Composite Number is a number that has factors.
A Prime Factor is a prime number used as a factor.
A Composite Factor is a composite number used as a factor.
Factoring is separating a number into its factors.
To find the Prime Factors of a Number.

RULE. Divide the number by any prime factor. Divide the quotient, if composite, in like manner; and so continue until a prime quotient is found. The several divisors and the last quotient will be the prime factors.

CANCELLATION Cancellation is rejecting equal factors from dividend and divisor.

PRINCIPLE. Dividing dividend and divisor by the same number does not affect the quotient.

GREATEST COMMON DIVISOR A Common Factor (divisor or measure) is a number that is a factor of each of two or more numbers. : A Common Prime Factor is a prime number that is a factor of each of two or more numbers.

The Greatest Common Factor (divisor or measure) is the largest number that is a factor of each of two or more numbers.

Numbers are prime to each other when they have no common factor.

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