Arithmetic

One object and one object make two objects; two objects and one object make three objects; three and one make four; and so on. One, two, three, four, and so on are called whole numbers.

The number one is also called a unit. The number two may be regarded as a collection of two units, the number three as a collection of three units, and so on. Thus every whole number is either a unit or a collection of several units.

Besides whole numbers, arithmetic also studies other kinds of numbers. We shall meet those later.

If to a unit we add another unit, and to the number obtained we again add a unit, and so on, we obtain the natural sequence of numbers: one, two, three, four, five, six, seven, and so on.

The smallest number in this sequence is one; there is no largest, because to any number, however large, we can always add another unit and obtain a still larger number. Thus the natural sequence can be continued without end; it is said to be infinite.

Of two different numbers, the one that comes earlier in the natural sequence is the smaller, and the one that comes later is the larger.

To find out how many desks there are in a classroom or how many trees in an orchard, we must count them. Counting consists in separating one object after another — actually or only in the mind — and naming at each step the number of objects separated. If at the separation of the last object we say, for example, eight, then there are eight objects; the number eight is in this case the result of the count.

We take it as an obvious truth that the result of counting does not depend on the order in which we count the objects. The important thing is only that in counting no object is missed and none is counted twice.

The first ten numbers of the natural sequence bear the following names: one, two, three, four, five, six, seven, eight, nine, ten.

With the help of these names and a few others, further numbers may also be expressed. Suppose we wish to name the number of strokes shown here:

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

We count off ten strokes and separate them from the rest, then count off another ten and separate them again. We continue counting by tens until either no strokes remain or fewer than ten remain. We then count the tens and the remaining strokes (units). If there are four tens and three remaining strokes, we name the total: four tens three units.

When a number contains more than ten tens, we count off ten tens, then another ten tens, and so on. Every ten tens is called by a single word: one hundred, or a hundred.

If a number contains three hundreds, five remaining tens, and seven remaining units, it may be named: three hundreds five tens seven units.

If there are more than ten hundreds, the hundreds too are counted in tens. Every ten hundreds is called one thousand.

In our language some shortened names of numbers are in common use. Thus ten-plus-one is called eleven, ten-plus-two is called twelve, and so on. Two tens are called twenty, three tens thirty, four tens forty. Two hundreds are called two hundred, three hundreds three hundred, and so on.

The first nine numbers are written by means of special signs called digits:

With the help of these nine digits and a tenth one, 0, called zero and denoting the absence of number, any number may be written.

For this purpose it is agreed that simple units are written in the first place from the right, tens in the second place from the right, hundreds in the third place. For example:

All digits except zero are called significant digits. These examples show why zero is needed: in writing three hundred forty we cannot omit the zero, because 34 would mean thirty-four. On the other hand, zeros on the left of the first significant digit may be omitted: 045 means the same as 45.

A number written with one digit is called a one-digit number; with two digits, a two-digit number; and so on.

When the count exceeds one thousand, we form as many thousands as possible, count the thousands and the remaining units, and name the number of both. For example: two hundred forty thousand five hundred sixty-two units.

One thousand thousands make one million; one thousand millions make one milliard (or billion); one thousand milliards make one trillion; and so on.

Suppose we wish to write the number thirty-five milliard eight hundred six million seven thousand sixty-three. It may be written using digits and words as: 35 milliard 806 million 7 thousand 63 units.

To do away with words entirely, it was agreed first to write the groups of milliards, millions, thousands, and simple units side by side in one row from left to right, and second always to write each such group with three digits. Thus instead of 63 units we write 063, instead of 7 thousand we write 007, and so on:

As before, leading zeros are omitted, giving: 35 806 007 063, or without spaces: 35806007063.

When reading back such a number, the first three digits from the right give the number of units, the next three the number of thousands, the next three the millions, and so on:

To read a number more easily, separate it mentally from the right into groups of three. For example 5183000567029 is read as: 5 trillion 183 milliard 567 thousand 29. Written with spaces — 5 183 000 567 029 — it can be read without mental grouping.

Each position has its own fixed meaning:

Our system of writing numbers rests on ten digits to which a double meaning is assigned: one meaning depends on the shape of the digit, the other on its position. Of two digits side by side, the left one denotes units ten times greater than the right one.

Units, tens, hundreds, thousands, and so on are sometimes called by another name: units are called units of the first order, tens units of the second order, hundreds units of the third order, and so on.

All units except simple units are called compound units. A ten, a hundred, and a thousand are compound units. Every compound unit contains ten units of the next lower order.

The orders of units are grouped into classes. The first class contains the first three orders: hundreds, tens, and units. The second class contains the next three orders: thousands, tens of thousands, and hundreds of thousands. The first class is the class of units; the second is the class of thousands.

To find how many hundreds are contained in the number 56,284, we note that simple hundreds occupy the third place from the right, where the digit is 2: two simple hundreds. The thousands digit (6) represents 6 thousands = 60 hundreds. The ten-thousands digit (5) represents 5 ten-thousands = 500 hundreds. Altogether: 500 + 60 + 2 = 562 hundreds.

Any general method of naming and writing numbers is called a numeration system. Our system is called decimal (or denary) because in it 10 units of one order make one unit of the next higher order. The number 10 is therefore called the base of the decimal system. Every number N is represented as:

Other systems can be imagined with a different base. With base 5 (the quinary system), 5 units of one order make one unit of the next, and every number takes the form:

where each of a, b, c, d, e, … is less than 5.

The decimal system uses ten symbols. The quinary system needs only five: 0, 1, 2, 3, 4. In it, the number 5 would be written 10 (one unit of the second order). For a system with base greater than 10, new symbols would have to be invented — in the duodecimal system (base 12), special signs would be needed for 10 and 11.

To express 1766 in the quinary system, divide successively by 5 and record the remainders:

To convert 5623₈ (octal) to decimal, use nested multiplication (Horner's method):

The Romans used only seven signs:

Roman digits retain their value regardless of position. When written side by side, their values are added: XXV = 10 + 10 + 5 = 25. There are six exceptions where a smaller digit before a larger digit is subtracted:

The units that make up several numbers can be combined into one collection. The number obtained by counting all those units is called the sum, and the numbers being combined are called addends. For instance: 5 matches and 7 matches and 2 matches combine into a collection of 14 matches. The number 14 is the sum of three addends: 5, 7, and 2.

The addends can be regarded as parts of the sum.

The operation that consists in forming the sum of several numbers is called addition. The sign of addition is + (plus). Addition is always possible and always gives a unique result.

1) Commutative law. The sum does not change when the order of the addends is changed:

In general: a + b + c = b + a + c = … for any numbers a, b, c.

2) Associative law. The sum does not change if any group of addends is replaced by their sum:

In general: a + b + c = (a + b) + c = a + (b + c).

1) To add the sum of several numbers to some number, add each addend one by one:

2) To add some number to a sum, add it to any one of the addends:

To find the sum of two single-digit numbers, add all the units of one to the other. To add all numbers quickly, memorise all sums arising from pairs of single-digit numbers.

To add 37 and 8: separate 7 units from 37, add them to 8 to get 15; add 15 to 30: since 15 = 10 + 5, adding 10 to 30 gives 40, and 40 + 5 = 45.

Alternatively: 37 needs 3 more to reach 40; separate 3 from 8; 37 + 3 = 40, and 5 remain: 40 + 5 = 45.

Write the addends with units under units, tens under tens, hundreds under hundreds, and so on. Draw a line below the last addend. Then add column by column from right to left, carrying whenever a column sum reaches 10 or more.

Adding units: 3+9+8+6 = 26 = 2 tens + 6 units; write 6, carry 2. Adding tens: 5+0+0+4+2 = 11 = 1 hundred + 1 ten; write 1, carry 1. And so on.

We now agree to treat zero as a number on equal footing with all others; zero is less than every other number.

To increase a number by several units means to add those units to it. Increasing a number by a certain amount is performed by addition.

If several units are added to one addend (others unchanged), the sum increases by that many units.
If several units are taken from one addend, the sum decreases by that many units.
If the same number of units is added to one addend and taken from another, the sum remains unchanged.

The operation of taking from one number as many units as another given number contains is called subtraction.

The number from which we subtract is the minuend; the number subtracted is the subtrahend; the result is the remainder or difference. The sign of subtraction is − (minus): 7 − 3 = 4.

The subtrahend cannot exceed the minuend.

In subtraction the minuend is broken into two numbers: subtrahend and remainder. Combining them gives back the minuend — so the minuend is the sum and the subtrahend and remainder are the addends. In addition, addends are given and the sum is sought; in subtraction, the sum and one addend are given and the other addend is sought. Subtraction is the inverse operation to addition.

Subtraction is always possible and gives a unique result provided the subtrahend is not greater than the minuend. If b = a, the remainder is zero; if b > a, subtraction is impossible (in the domain of whole numbers).

The required difference is easily found by using addition. For example: 15 − 8 = ? — recall that 8 + 7 = 15, so 15 − 8 = 7.

Write the subtrahend below the minuend, aligning by place. Subtract column by column from right to left. When a digit of the minuend is too small, borrow 1 from the next higher order.

Units: can't subtract 5 from 2; borrow from tens, making 12 − 5 = 7. Tens: only 6 remain after borrowing; 6 − 4 = 2. Hundreds: none in minuend, must borrow from ten-thousands, then thousands. After all borrowing: 7 hundreds. Thousands: 9 − 7 = 2. Ten-thousands: 5 − 0 = 5. Result: 52727.

1) To subtract a sum, subtract each addend separately: a − (b + c + …) = a − b − c − …

2) To subtract a number from a sum, subtract it from any one addend: (a + b + c) − m = (a − m) + b + c

Add the addends again in a different order (e.g., bottom to top). If the same sum results, the addition is very probably correct. Alternatively, subtract one addend from the sum; the result should equal the sum of the remaining addends.

Add the subtrahend and the remainder; the result should equal the minuend. Alternatively, subtract the remainder from the minuend; the result should equal the subtrahend.

To decrease a number by several units means to subtract those units from it. This is performed by subtraction.

To find by how much one number exceeds another, subtract the smaller from the larger. For example: 35 − 20 = 15, so 35 is greater than 20 by 15.

Adding to the minuend increases the remainder by the same amount.
Subtracting from the minuend decreases the remainder by the same amount.
Adding to the subtrahend decreases the remainder by the same amount.
Subtracting from the subtrahend increases the remainder by the same amount.

Notably: the remainder does not change if both minuend and subtrahend are increased or decreased by the same number simultaneously: (11 + 6) − (3 + 6) = 11 − 3 = 8.

To subtract (12 − 8) from 30: increase both the 30 and the subtrahend (12 − 8) by 8, giving 38 − 12 = 26. Or: 30 − 12 = 18, then add back 8: 18 + 8 = 26.

To indicate operations without performing them, write the numbers with the appropriate signs between them:

10 + 15 + 20 (read: "ten plus fifteen plus twenty" or "the sum of 10, 15, 20")
10 − 8 (read: "ten minus eight" or "the difference of 10 and 8")

Other signs in common use: = (equals), > (greater than), < (less than), ≠ (not equal to), ≤ (less than or equal to), ≥ (greater than or equal to).

The signs > and < must have their point aimed at the smaller number.

Brackets show which operations must be performed first. For example: 200 − (35 + 20) means subtract the sum 35 + 20 from 200. Nested brackets use different shapes:

means: compute 7+8=15; subtract from 60 to get 45; subtract from 160 to get 115; add to 100 to get 215. The innermost brackets are always evaluated first.

When no brackets are present: in an expression showing only additions and subtractions, carry out operations left to right. So 20 − 2 + 4 − 5 means ((20 − 2) + 4) − 5 = 17.

An expression showing what operations are to be performed on given numbers and in what order, to obtain a required result, is called a formula. To evaluate a formula means to find the number resulting from all the indicated operations.

Multiplication is the addition of equal addends.

The number repeated as an addend is the multiplicand; the number showing how many such addends are taken is the multiplier; the result is the product. Both are called factors.

The multiplication sign is · (dot) or × (cross): 85·6 = 510. When factors are letters, no sign is needed: ab means a times b.

1·5 = 5 (multiplicand = 1: product equals multiplier).
0·4 = 0 (multiplicand = 0: product is zero).
5·1 = 5 (multiplier = 1: product equals multiplicand).
5·0 = 0 (multiplier = 0: product is zero).

To increase a number 2, 3, 4 … times means to form the sum of that many equal addends, each equal to the given number. This is performed by multiplication. (Compare: increasing by a number is addition; increasing by a factor is multiplication.)

Consider a rectangular array of strokes — 7 rows of 3. Counting by rows gives 7·3; counting by columns gives 3·7. The total is the same. In general: a·b = b·a (commutative law of multiplication).

To multiply quickly, memorise all products of pairs of single-digit numbers — the multiplication table. See the interactive table above.

To multiply 846 by 5: multiply units, then tens, then hundreds, carrying as needed.

Multiply by each digit of the multiplier separately (partial products), placing each one shifted one position left for each successive digit. Then add all partial products.

Ignore the trailing zeros, multiply the significant parts, then append the total count of trailing zeros:

If one factor is multiplied by n, the product is multiplied by n.
If one factor is divided by n, the product is divided by n.
If one factor is multiplied by n and the other divided by n, the product is unchanged.

The product 3·4·2·7 means ((3·4)·2)·7. The result is the same regardless of grouping or order of factors.

2·5·3·4·7 = 2·3·4·5·7 = 4·7·3·2·5 = 840 in all orderings (commutative law, extended to any number of factors).

3·4·(5·2) = 12·10 = 120 = 3·4·5·2 (associative law of multiplication): abc = (ab)c = a(bc).

The operation of finding one of two factors when the product and the other factor are given is called division.

The given product is the dividend; the given factor is the divisor; the unknown factor is the quotient.

Division is denoted ÷ or by a horizontal fraction bar: 75 ÷ 3 = 25.

If the dividend is not zero, dividing by zero would require finding a number that, when multiplied by zero, gives the dividend — but no such number exists. If the dividend is also zero, the quotient could be any number at all, so the result is not unique. Zero cannot serve as a divisor.

When exact division is impossible (27 ÷ 6 has no whole-number solution), we speak of division with a remainder. The incomplete quotient is the largest whole number q such that q × (divisor) ≤ dividend. The remainder is the difference. The remainder is always less than the divisor.

To divide a by b (b ≠ 0) means to find whole numbers q (quotient) and r (remainder) such that:

This pair (q, r) always exists and is unique. The quotient shows how many times the divisor fits into the dividend. The remainder can be any value from 0 to b − 1; there are b distinct possible remainders when dividing by b.

In multiplication, two factors are given and the product is found. In division, the product and one factor are given and the other factor is sought. Division is the inverse operation to multiplication.

Multiply the divisor by 10 and compare with the dividend. If the dividend is smaller, the quotient is single-digit; otherwise it is not.

Divide 64528 by 23. Think of it as distributing 64528 equally into 23 parts.

To divide by a product, divide successively by each factor: 60 ÷ (5·3) = 60 ÷ 5 ÷ 3 = 12 ÷ 3 = 4.

Multiply the dividend by n → quotient multiplied by n.
Divide the dividend by n → quotient divided by n.
Multiply the divisor by n → quotient divided by n.
Divide the divisor by n → quotient multiplied by n.
Multiply both by n, or divide both by n → quotient unchanged.

1) To divide a sum, divide each addend separately (assuming exact division): (21 + 14 + 35) ÷ 7 = 3 + 2 + 5 = 10.

2) To divide a difference, divide minuend and subtrahend separately: (40 − 25) ÷ 5 = 8 − 5 = 3.

Addition and subtraction are called operations of the first level; multiplication and division are operations of the second level. When no brackets are present: perform second-level operations before first-level operations. For example: 2 + 3·4 − 6÷2 = 2 + 12 − 3 = 11.

Of the four arithmetic operations, two — addition and multiplication — can always be performed (on any numbers). Subtraction is not always possible, but the test is simple: the minuend must not be smaller than the subtrahend. Division is more subtle: it often cannot be carried out exactly, and it may not be at all obvious whether one number divides another without first actually dividing. For this reason, the hardest questions in arithmetic are bound up with division. We shall study some of them in this part.

When one number divides another without remainder, we simply say the first number is divisible by the second. In this case the first is also called a multiple of the second, and the second is called a divisor of the first. So 15 is a multiple of 3, and 3 is a divisor of 15.

Zero is divisible by any non-zero number, and the quotient is then zero.

We frequently use these properties when establishing divisibility tests:

Numbers divisible by 2 are called even; the rest are odd. Odd and even numbers alternate in the natural sequence.

Any number ending in 0 is a sum of tens. Every ten is divisible by 2, so any number ending in 0 is divisible by 2. If a number ends in any even digit (2, 4, 6, 8), write it as a multiple of 10 plus that digit: both parts are divisible by 2, so the sum is also. If it ends in an odd digit, the sum is not divisible by 2.

Every hundred is divisible by 4 (100 = 4 × 25). So any number ending in two zeros is divisible by 4.

Every number formed by repeating the digit 9 (9, 99, 999, …) is divisible by both 3 and 9. Every thousand = 999 + 1, every hundred = 99 + 1, every ten = 9 + 1. Splitting a number into its individual digits:

So divisibility by 3 (or 9) depends entirely on the digit sum.

The justification for the 6-rule: if a number breaks into 6's, then the 6's break into 2's and also into 3's, so it is divisible by both 2 and 3. Conversely, if it is divisible by both 2 and 3, write it as 3·k where k is even (since the number is divisible by 2, and 3 is odd, k must be even), so 3·k = 3·2·m = 6·m.

Every whole number greater than 1 has at least two divisors: 1 and itself. Numbers that have exactly two divisors are called prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 …

Numbers with more than two divisors are called composite: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …

The unit (1) is neither prime nor composite.

Suppose there were finitely many primes: p₁, p₂, …, pₙ. Form the number N = p₁·p₂·…·pₙ + 1. It leaves remainder 1 when divided by any of the known primes, so none of them divides N. But N is either prime (giving a new prime) or composite (and must have a prime divisor not in our list). Either way the list was incomplete — a contradiction. So primes are infinite in number.

If a number N has no prime divisors ≤ √N, it is prime.

A number that is a divisor of each of several given numbers is called a common divisor. The largest such number is the greatest common divisor (GCD).

To find the GCD of 12 and 18: prime factors of 12 = 2²·3, prime factors of 18 = 2·3². Common factors: 2¹·3¹ = 6. GCD(12,18) = 6.

Divide the larger number by the smaller; then divide the previous divisor by the remainder; continue until the remainder is zero. The last nonzero remainder is the GCD.

A number that is a multiple of each of several given numbers is a common multiple. The smallest positive such number is the least common multiple (LCM).

Until now we have dealt only with whole numbers. The historical origin of whole numbers was above all the need to count, and whole numbers satisfy that need completely. But human activity since antiquity has generated needs that whole numbers cannot meet. The necessity arose to introduce new numbers. One of the fundamental activities that made this necessary is the measurement of quantities. We therefore pause to examine what measurement is before proceeding to study these new numbers.

To measure the length of a room, we use some familiar unit of length — say, a metre — and count how many times it fits along the room. If it fits exactly 10 times, the room is 10 metres long. Similarly, to measure a weight, we use a unit of weight (e.g., a gram) and count how many times it fits.

A metre is the unit of length; a gram is the unit of weight; and so on. For each kind of quantity several units may be used — larger ones and smaller ones. The gram is used alongside the kilogram, tonne, and milligram.

The metric system is now in use in the USSR and in many other countries. Its unit of length is the metre. Submultiples are formed with Latin prefixes: deci- (tenth), centi- (hundredth), milli- (thousandth). Multiples use Greek prefixes: deca- (×10), hecto- (×100), kilo- (×1000).

Numbers that show the measurement of some quantity are called denominate numbers. When working with them, always convert to a common unit before adding or subtracting.

One centimetre is the hundredth part of a metre; one hour is the twenty-fourth part of a day. We call a centimetre the "hundredth fraction" of a metre, and an hour the "twenty-fourth fraction" of a day. A minute is the sixtieth fraction of an hour. The word fraction replaces "part" in this context (the Russian word доля is more precise, meaning a measured portion).

The second fraction is commonly called a half, the third a third, the fourth a quarter.

One fraction, or a collection of several equal fractions of a unit, is called a fraction. Examples: one tenth, three fifths, twelve sevenths. A whole number together with a fraction forms a mixed number: "3 and seven eighths". Fractions and mixed numbers are called fractional numbers, in contrast to whole numbers.

Write the number of fractions above a horizontal line; below the line write how many equal parts the unit was divided into.

The number above the line is the numerator (how many fractions are taken); below is the denominator (how many parts the unit was divided into). Both together are the terms of the fraction.

A mixed number is written as a whole number with the fraction written to its right: 327.

Divide 5 kg of bread into 8 equal parts. Imagine each kilogram divided into 8 equal parts: 5 kg contains 5×8=40 such parts. One eighth of 5 kg is 40÷8=5 of those parts. So one eighth of 5 kg is 58 kg.

A fraction is proper when the numerator is smaller than the denominator (35, 712): its value is less than 1. An improper fraction has numerator ≥ denominator (75, 93): its value is ≥ 1.

Every improper fraction can be converted to a whole number or mixed number by dividing numerator by denominator: 175 = 325; conversely, 325 = 175.

When the denominator is the same, the fraction with the larger numerator is larger: 59 > 39.

To reduce a fraction means to divide both terms by a common divisor. A fraction is in its lowest terms when the GCD of numerator and denominator is 1 (they are coprime). To reduce to lowest terms, divide both terms by their GCD.

To compare or add fractions with different denominators, convert them to have the same denominator — the least common denominator (LCD) = LCM of the original denominators.

Cross-cancellation before multiplying saves work: if any numerator and any denominator share a common factor, cancel it first.

The fractions obtained by dividing a unit into 10, 100, 1000, … equal parts are called decimal fractions:

Each decimal fraction of one order contains 10 decimal fractions of the next lower order.

A fraction whose denominator is 1 followed by zeros is called a decimal fraction: 310, 27100, 274011000. Fractions with any other denominator are called common or vulgar fractions.

In a whole number, each digit to the left represents units 10 times greater than the digit to its right. We extend this convention to the right of the simple units. We separate the whole part from the decimal fractions by a comma (in this text) or a dot.

Digits to the right of the comma are called decimal places.

This follows from the rule on multiplying/dividing fraction terms: 0,7 = 710 = 70100 = 0,70.

First compare whole parts; the one with the larger whole part is greater. If whole parts are equal, compare tenths; if those are equal, compare hundredths; and so on. To facilitate comparison, equalize the number of decimal places by appending zeros:

A fraction whose denominator (in lowest terms) has no prime factors other than 2 and 5 produces a terminating decimal. Any other fraction produces a recurring decimal — one with a repeating block (the period).

A proportion is an equality of two ratios:

Every proportion has two antecedents and two consequents. In the proportion a:b = c:d, the outer terms a and d are called the extremes; the inner terms b and c are the means.

This follows directly by cross-multiplying both sides by bd.

From one proportion, four valid proportions can be derived. If ab = cd, then also: ba = dc  and  ac = bd  and  ca = db.

If three terms are known, the fourth is found from a·d = b·c:

Two quantities are in direct proportion when an increase or decrease of one by a factor causes the same change in the other. Their ratio remains constant.

Two quantities are in inverse proportion when an increase in one causes a proportional decrease in the other, their product remaining constant.

To divide a number in a given ratio: find the sum of the ratio terms; divide the number by this sum to find the unit share; then multiply each ratio term by the unit share.

A per cent is one hundredth part of a quantity (from Latin per centum). The symbol is %. 1% = 1100 = 0,01.