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English number words include numerals and various words derived from them, as well as a large number of words borrowed from other languages.
Cardinal numbers
Cardinal numbers refer to the size of a group. In English, these words are numerals.
| 0 | zero (nought) | 10 | ten | ||
| 1 | one | 11 | eleven | ||
| 2 | two | 12 | twelve (a dozen) | 20 | twenty (a score) |
| 3 | three | 13 | thirteen | 30 | thirty |
| 4 | four | 14 | fourteen | 40 | forty (no "u") |
| 5 | five | 15 | fifteen (note "f", not "v") | 50 | fifty (note "f", not "v") |
| 6 | six | 16 | sixteen | 60 | sixty |
| 7 | seven | 17 | seventeen | 70 | seventy |
| 8 | eight | 18 | eighteen (only one "t") | 80 | eighty (only one "t") |
| 9 | nine | 19 | nineteen | 90 | ninety (note the "e") |
If a number is in the range 21 to 99, and the second digit is not zero, one typically writes the number as two words separated by a hyphen.
| 21 | twenty-one |
| 25 | twenty-five |
| 32 | thirty-two |
| 58 | fifty-eight |
| 64 | sixty-four |
| 79 | seventy-nine |
| 83 | eighty-three |
| 99 | ninety-nine |
In English, the hundreds are perfectly regular, except that the word hundred remains in its singular form regardless of the number preceding it.
| 100 | one hundred |
| 200 | two hundred |
| … | … |
| 900 | nine hundred |
So too are the thousands, with the number of thousands followed by the word "thousand"
| 1,000 | one thousand |
| 2,000 | two thousand |
| … | … |
| 10,000 | ten thousand or (rarely used) a myriad |
| 11,000 | eleven thousand |
| … | … |
| 20,000 | twenty thousand |
| 21,000 | twenty-one thousand |
| 30,000 | thirty thousand |
| 85,000 | eighty-five thousand |
| 100,000 | one hundred thousand or one lakh (Indian English) |
| 999,000 | nine hundred and ninety-nine thousand (inclusively British English, Irish English, Australian English, and New Zealand English) nine hundred ninety-nine thousand (American English) |
| 1,000,000 | one million |
| 10,000,000 | ten million or one crore (Indian English) |
In American usage, four-digit numbers with non-zero hundreds are often named using multiples of "hundred" and combined with tens and ones: "One thousand one", "Eleven hundred three", "Twelve hundred twenty-five", "Four thousand forty-two", or "Ninety-nine hundred ninety-nine." In British usage, this style is common for multiples of 100 between 1,000 and 2,000 (e.g. 1,500 as "fifteen hundred") but not for higher numbers.
Americans may pronounce four-digit numbers with non-zero tens and ones as pairs of two-digit numbers without saying "hundred" and inserting "oh" for zero tens: "twenty-six fifty-nine" or "forty-one oh five". This usage probably evolved from the distinctive usage for years; "nineteen-eighty-one", or from four-digit numbers used in the American telephone numbering system which were originally two letters followed by a number followed by a four-digit number, later by a three-digit number followed by the four-digit number. It is avoided for numbers less than 2500 if the context may mean confusion with time of day: "ten ten" or "twelve oh four".
Intermediate numbers are read differently depending on their use. Their typical naming occurs when the numbers are used for counting. Another way is for when they are used as labels. The second column method is used much more often in American English than British English. The third column is used in British English, but rarely in American English (although the use of the second and third columns is not necessarily directly interchangeable between the two regional variants). In other words, British English and American English can seemingly agree, but it depends on a specific situation (in this example, bus numbers).
| Common British vernacular | Common American vernacular | Common British vernacular | |
| "How many marbles do you have?" | "What is your house number?" | "Which bus goes to the high street?" | |
| 101 | "A hundred and one." | "One-oh-one." Here, "oh" is used for the digit zero. |
"One-oh-one." |
| 109 | "A hundred and nine." | "One-oh-nine." | "One-oh-nine." |
| 110 | "A hundred and ten." | "One-ten." | "One-one-oh." |
| 117 | "A hundred and seventeen." | "One-seventeen." | "One-one-seven." |
| 120 | "A hundred and twenty." | "One-twenty." | "One-two-oh", "One-two-zero." |
| 152 | "A hundred and fifty-two." | "One-fifty-two." | "One-five-two." |
| 208 | "Two hundred and eight." | "Two-oh-eight." | "Two-oh-eight." |
| 334 | "Three hundred and thirty-four." | "Three-thirty-four." | "Three-three-four." |
Note: When writing a cheque (or check), the number 100 is always written "one hundred". It is never "a hundred".
Note that in American English, many students are taught not to use the word and anywhere in the whole part of a number, so it is not used before the tens and ones. It is instead used as a verbal delimiter when dealing with compound numbers. Thus, instead of "three hundred and seventy-three", one would say "three hundred seventy-three". For details, see American and British English differences.
For numbers above a million, there are two different systems for naming numbers in English (for the use of prefixes such as kilo- for a thousand, mega- for a million, milli- for a thousandth, etc. see SI units):
- the long scale (decreasingly used in British English) designates a system of numeric names in which a thousand million is called a ‘‘milliard’’ (but the latter usage is now rare), and ‘‘billion’’ is used for a million million.
- the short scale (always used in American English and increasingly in British English) designates a system of numeric names in which a thousand million is called a ‘‘billion’’, and the word ‘‘milliard’’ is not used.
| Number notation | Power notation |
Short scale | Long scale | Indian (or South Asian) English |
|---|---|---|---|---|
| 1,000,000 | 106 | one million | one million | ten lakh |
| 1,000,000,000 | 109 | one billion a thousand million |
one milliard a thousand million |
one hundred crore (one arab) |
| 1,000,000,000,000 | 1012 | one trillion a thousand billion |
one billion a million million |
one lakh crore (ten kharab) |
| 1,000,000,000,000,000 | 1015 | one quadrillion a thousand trillion |
one billiard a thousand billion |
ten crore crore (one padm) |
| 1,000,000,000,000,000,000 | 1018 | one quintillion a thousand quadrillion |
one trillion a million billion |
ten thousand crore crore (ten shankh) |
| 1,000,000,000,000,000,000,000 | 1021 | one sextillion a thousand quintillion |
one trilliard a thousand trillion |
one crore crore crore |
The numbers past a trillion in short scale system, in ascending powers of ten, are as follows: quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quattuordecillion, and quindecillion (that's 10 to the 48th, or a one followed by 48 zeros). The highest number listed on Robert Munafo's table,[1] is a milli-millillion. That's 10 to the 3000003rd.
The googolplex has often been nominated as the largest named number in the world. If a googol is ten to the one hundredth, then a googolplex is one followed by a googol of zeroes.[2]
Although British English has traditionally followed the long-scale numbering system, the short-scale usage has become increasingly common in recent years. For example, the UK Government and BBC websites use the newer short-scale values exclusively.
The terms arab, kharab, padm and shankh are more commonly found in old sections of Indian Mathematics.
Here are some approximate composite large numbers in American English:
| Quantity | Written | Pronounced |
|---|---|---|
| 1,200,000 | 1.2 million | one point two million |
| 3,000,000 | 3 million | three million |
| 250,000,000 | 250 million | two hundred fifty million |
| 6,400,000,000 | 6.4 billion | six point four billion |
| 23,380,000,000 | 23.38 billion | twenty-three point three eight billion |
Often, large numbers are written with (preferably non-breaking) half-spaces or thin spaces separating the thousands (and, sometimes, with normal spaces or apostrophes) instead of commas—to ensure that confusion is not caused in countries where a decimal comma is used. Thus, a million is often written 1 000 000.
In some areas, a point (. or ·) may also be used as a thousands' separator, but then, the decimal separator must be a comma.
Specialized numbers
A few numbers have special names (in addition to their regular names):
- 0: has several other names, depending on context:
- zero: formal scientific usage
- naught / nought: mostly British usage
- aught: Mostly archaic but still occasionally used when a digit in mid-number is 0 (as in "thirty-aught-six", the .30-06 Springfield rifle cartridge and by association guns that fire it)
- oh: used when spelling numbers (like telephone, bank account, bus line [British: bus route])
- nil: in general sport scores, British usage ("The score is two–nil.")
- nothing: in general sport scores, American usage ("The score is two–nothing.")
- null: used technically to refer to an object or idea related to nothingness. The 0th aleph number ( ) is pronounced "aleph-null".
- love: in tennis, badminton, squash and similar sports (origin disputed, often said to come from French l'œuf, "egg"; but the Oxford English Dictionary mentions the phrase for love, meaning nothing is at risk)
- zilch, nada (from Spanish), zip: used informally when stressing nothingness; this is true especially in combination with one another ("You know nothing—zero, zip, nada, zilch!"); American usage
- nix: also used as a verb; mostly American usage
- cypher / cipher: archaic, from French chiffre, in turn from Arabic sifr, meaning zero
- goose egg (informal)
- duck (used in cricket when a batsman is dismissed without scoring)
- blank the half of a domino tile with no pips
- 1:
- ace in certain sports and games, as in tennis or golf, indicating success with one stroke, and the face of a die, playing card or domino half with one pip
- birdie in golf denotes one stroke less than par, and bogey, one stroke more than par
- solo
- unit
- linear the degree of a polynomial is 1; also for explicitly denoting the first power of a unit: linear meter
- unity in mathematics
- 2:
- couple
- brace, from Old French "arms" (the plural of arm), as in "what can be held in two arms".
- pair
- deuce the face of a die, playing card or domino half with two pips
- eagle in golf denotes two strokes less than par
- duo
- quadratic the degree of a polynomial is 2
- also square or squared for denoting the second power of a unit: square meter or meter squared
- 3:
- trey the face of a die or playing card with three pips, a three-point field goal in basketball, nickname for the third carrier of the same personal name in a family
- trio
- trips: three-of-a-kind in a poker hand. a player has three cards with the same numerical value
- cubic the degree of a polynomial is 3
- also cube or cubed for denoting the third power of a unit: cubic meter or meter cubed
- albatross in golf denotes three strokes less than par. Sometimes called double eagle
- hat-trick or hat trick: achievement of three feats in sport or other contexts[3]
- 4:
- cater: (rare) the face of a die or playing card with four pips
- quartet
- quartic or biquadratic the degree of a polynomial is 4
- quad (short for quadruple or the like) several specialized sets of four, such as four of a kind in poker, a carburetor with four inputs, etc.,
- condor in golf denotes four strokes less than par
- 5:
- cinque: (rare) the face of a die or playing card with five pips
- quintet
- nickel (informal American, from the value of the five-cent US nickel, but applied in non-monetary references)
- quintic the degree of a polynomial is 5
- quint (short for quintuplet or the like) several specialized sets of five, such as quintuplets, etc.
- 6:
- 7:
- 8:
- 9:
- 10:
- 11: a banker's dozen
- 12: a dozen (first power of the duodecimal base), used mostly in commerce
- 13: a baker's dozen
- 20: a score (first power of the vigesimal base), nowadays archaic; famously used in the opening of the Gettysburg Address: "Four score and seven years ago..." The Number of the Beast in the King James Bible is rendered "Six hundred threescore and six".
- 50: half a century, literally half of a hundred, usually used in cricket scores.
- 60: a shock: historical commercial count, described as "three scores".[4]
- 100:
- 110: eleventy (as 11 × 10)
- 120:
- A great hundred or long hundred (twelve tens; as opposed to the small hundred, i.e. 100 or ten tens), also called small gross (ten dozens), both archaic
- Also sometimes referred to as duodecimal hundred, although that could literally also mean 144, which is twelve squared
- Twelfty or twelvety (as 12 × 10)
- 144: a gross (a dozen dozens, second power of the duodecimal base), used mostly in commerce
- 1000:
- a grand, colloquially used especially when referring to money, also in fractions and multiples, e.g. half a grand, two grand, etc. Grand can also be shortened to "G" in many cases.
- K, originally from the abbreviation of kilo-, e.g. "He only makes $20K a year."
- 1728: a great gross (a dozen gross, third power of the duodecimal base), used mostly in commerce
- 10,000: a myriad (a hundred hundred), commonly used in the sense of an indefinite very high number
- 100,000: a lakh (a hundred thousand), loanword used mainly in Indian English
- 10,000,000: a crore (a hundred lakh), loanword used mainly in Indian English
- 10100: googol (1 followed by 100 zeros), used in mathematics; not to be confused with the name of the company Google (which was originally a misspelling of googol)
- 10googol: googolplex (1 followed by a googol of zeros)
- 10googolplex: googolplexplex (1 followed by a googolplex of zeros)
Combinations of numbers in most sports scores are read as in the following examples:
- 1–0 British English: one-nil; American English: one-nothing, one-zip, or one-zero
- 0–0 British English: nil-nil, or more rarely nil all; American English: zero-zero or nothing-nothing, (occasionally scoreless or no score)
- 2–2 two-two or two all; American English also twos, two to two, even at two, or two up.
Naming conventions of Tennis scores (and related sports) are different from other sports.
Multiplicative adverbs
A few numbers have specialised multiplicative numbers expresses how many times some event happens (adverbs):
| one time | once |
| two times | twice |
| three times | thrice |
Compare these specialist multiplicative numbers to express how many times some thing exists (adjectives):
| × 1 | solitary | singular | one-off |
| × 2 | double | twofold | duplicate |
| × 3 | triple | threefold | triplicate |
| × 4 | quadruple | fourfold | |
| × 5 | quintuple | fivefold | |
| × 6 | sextuple | sixfold | |
| × 7 | septuple | sevenfold | |
| ×100 | ..... | hundredfold |
English also has some multipliers and distributive numbers, such as singly.
Other examples are given in the Specialist Numbers.
Negative numbers
The name of a negative number is the name of the corresponding positive number preceded by "minus" or (American English) "negative". Thus −5.2 is "minus five point two" or "negative five point two". For temperatures, Americans colloquially say "below" — short for "below zero" — so a temperature of −5° is "five below" (in contrast, for example, to "two above" for 2°, occasionally used for emphasis when referring to several temperatures or ranges both positive and negative).
Ordinal numbers
Ordinal numbers refer to a position in a series. Common ordinals include:
| 0th | zeroth or noughth (see below) | 10th | tenth | ||
| 1st | first | 11th | eleventh | ||
| 2nd | second | 12th | twelfth (note "f", not "v") | 20th | twentieth |
| 3rd | third | 13th | thirteenth | 30th | thirtieth |
| 4th | fourth | 14th | fourteenth | 40th | fortieth |
| 5th | fifth | 15th | fifteenth | 50th | fiftieth |
| 6th | sixth | 16th | sixteenth | 60th | sixtieth |
| 7th | seventh | 17th | seventeenth | 70th | seventieth |
| 8th | eighth (only one "t") | 18th | eighteenth | 80th | eightieth |
| 9th | ninth (no "e") | 19th | nineteenth | 90th | ninetieth |
Zeroth only has a meaning when counting starts with zero, which happens in a mathematical or computer science context.
Ordinal numbers such as 21st, 33rd, etc., are formed by combining a cardinal ten with an ordinal unit.
| 21st | twenty-first |
| 25th | twenty-fifth |
| 32nd | thirty-second |
| 58th | fifty-eighth |
| 64th | sixty-fourth |
| 79th | seventy-ninth |
| 83rd | eighty-third |
| 99th | ninety-ninth |
Higher ordinals are not often written in words, unless they are round numbers (thousandth, millionth, billionth). They are written using digits and letters as described below. Here are some rules that should be borne in mind.
- The suffixes -th, -st, -nd and -rd are occasionally written superscript above the number itself.
- If the tens digit of a number is 1, then write "th" after the number. For example: 13th, 19th, 112th, 9,311th.
- If the tens digit is not equal to 1, then use the following table:
| If the units digit is: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| write this after the number | th | st | nd | rd | th | th | th | th | th | th |
- For example: 2nd, 7th, 20th, 23rd, 52nd, 135th, 301st.
These ordinal abbreviations are actually hybrid contractions of a numeral and a word. 1st is "1" + "st" from "first". Similarly, "nd" is used for "second" and "rd" for "third". In the legal field and in some older publications, the ordinal abbreviation for "second" and "third" is simply "d".
- For example: 42d, 33d, 23d.
NB: The practice of using "d" to denote "second" and "third" is still often followed in the numeric designations of units in the US armed forces, for example, 533d Squadron.
Any ordinal name that doesn't end in "first", "second", or "third", ends in "th".
Dates
There are a number of ways to read years. The following table offers a list of valid pronunciations and alternate pronunciations for any given year of the Gregorian calendar.
| Year | Most common pronunciation method | Alternative methods |
|---|---|---|
| 1 BC | (The year) One Before Christ (BC) | 1 before the Common era (BCE) |
| 1 | (The year) One Anno Domini (AD) | of the Common era (CE) In the year of Our Lord 1 |
| 235 | Two thirty-five | Two-three-five Two hundred (and) thirty-five |
| 911 | Nine eleven | Nine-one-one Nine hundred (and) eleven |
| 999 | Nine ninety-nine | Nine-nine-nine Nine hundred (and) ninety-nine |
| 1000 | One thousand | Ten hundred 1K |
| 1004 | One thousand (and) four | Ten oh-four |
| 1010 | Ten ten | One thousand (and) ten |
| 1050 | Ten fifty | One thousand (and) fifty |
| 1225 | Twelve twenty-five | One-two-two-five One thousand, two hundred (and) twenty-five Twelve-two-five |
| 1900 | Nineteen hundred | One thousand, nine hundred Nineteen aught |
| 1901 | Nineteen oh-one | Nineteen hundred (and) one One thousand, nine hundred (and) one Nineteen aught one |
| 1919 | Nineteen nineteen | Nineteen hundred (and) nineteen One thousand, nine hundred (and) nineteen |
| 1999 | Nineteen ninety-nine | Nineteen hundred (and) ninety-nine One thousand, nine hundred (and) ninety-nine |
| 2000 | Two thousand | Twenty hundred Two triple-oh Y2K |
| 2001 | Two thousand (and) one | Twenty oh-one Twenty hundred (and) one Two double-oh-one Two oh-oh-one |
| 2009 | Two thousand (and) nine | Twenty oh-nine Twenty hundred (and) nine Two double-oh-nine Two oh-oh-nine |
| 2010 | Two thousand (and) ten Twenty ten [6] |
Twenty hundred (and) ten two-oh-one-oh |
Fractions and decimals
In spoken English, ordinal numbers are also used to quantify the denominator of a fraction. Thus "fifth" can mean the element between fourth and sixth, or the fraction created by dividing the unit into five pieces. In this usage, the ordinal numbers can be pluralized: one seventh, two sevenths. The sole exception to this rule is division by two. The ordinal term "second" can only refer to location in a series; for fractions English speakers use the term 'half' (plural "halves").
Here are some common English fractions (known linguistically as "partitive numerals"):[7]
| 1/16 | one sixteenth |
| 1/10 or 0.1 | one tenth |
| 1/8 | one eighth |
| 2/10 or 0.2 | two tenths |
| 1/4 | one quarter or (mainly American English) one fourth |
| 3/10 or 0.3 | three tenths |
| 1/3 | one third |
| 3/8 | three eighths |
| 4/10 or 0.4 | four tenths |
| 1/2 | one half |
| 6/10 or 0.6 | six tenths |
| 5/8 | five eighths |
| 2/3 | two thirds |
| 7/10 or 0.7 | seven tenths |
| 3/4 | three quarters or three fourths |
| 8/10 or 0.8 | eight tenths |
| 7/8 | seven eighths |
| 9/10 or 0.9 | nine tenths |
| 15/16 | fifteen sixteenths |
Alternatively, and for greater numbers, one may say for 1/2 "one over two", for 5/8 "five over eight", and so on. This "over" form is also widely used in mathematics.
Numbers with a decimal point may be read as a cardinal number, then "and", then another cardinal number followed by an indication of the significance of the second cardinal number (mainly U.S.); or as a cardinal number, followed by "point", and then by the digits of the fractional part. The indication of significance takes the form of the denominator of the fraction indicating division by the smallest power of ten larger than the second cardinal. This is modified when the first cardinal is zero, in which case neither the zero nor the "and" is pronounced, but the zero is optional in the "point" form of the fraction.
For example:
- 0.002 is "two thousandths" (mainly U.S.); or "point zero zero two", "point oh oh two", "nought point zero zero two", etc.
- 3.1416 is "three point one four one six"
- 99.3 is "ninety-nine and three tenths" (mainly U.S.); or "ninety-nine point three".
In English the decimal point was originally printed in the center of the line (0·002), but with the advent of the typewriter it was placed at the bottom of the line, so that a single key could be used as a full stop/period and as a decimal point. In many non-English languages a full-stop/period at the bottom of the line is used as a thousands separator with a comma being used as the decimal point.
Fractions together with an integer are read as follows:
- 1 1/2 is "one and a half"
- 6 1/4 is "six and a quarter"
- 7 5/8 is "seven and five eighths"
A space is required between the whole number and the fraction; however, if a special fraction character is used like "½", then the space can be done without, e.g.
- 9 1/2
- 9½
Whether to use digits or words
With very little deviation, most grammatical texts rule that the numbers zero to nine inclusive should be "written out" – meaning instead of "1" and "2", one would write "one" and "two".[8]
- Example: "I have two apples." (Preferred)
- Example: "I have 2 apples."
After "nine", one can head straight back into the 10, 11, 12, etc., although some write out the numbers until "twelve".
- Example: "I have 28 grapes." (Preferred)
- Example: "I have twenty-eight grapes."
Another common usage is to write out any number that can be expressed as one or two words, and use figures otherwise.
- Examples:
- "There are six million dogs." (Preferred)
- "There are 6,000,000 dogs."
- "That is one hundred and twenty-five oranges." (British English)
- "That is one hundred twenty-five oranges." (US-American English)
- "That is 125 oranges." (Preferred)
Numbers at the beginning of a sentence should also be written out.
The above rules are not always used. In literature, larger numbers might be spelled out. On the other hand, digits might be more commonly used in technical or financial articles, where many figures are discussed. In particular, the two different forms should not be used for figures that serve the same purpose; for example, it is inelegant to write, "Between day twelve and day 15 of the study, the population doubled."
Empty numbers
Colloquial English has a small vocabulary of empty numbers that can be employed when there is uncertainty as to the precise number to use, but it is desirable to define a general range: specifically, the terms "umpteen", "umpty", and "zillion". These are derived etymologically from the range affixes:
- "-teen" (designating the range as being between 10 and 20)
- "-ty" (designating the range as being in one of the decades between 20 and 100)
- "-illion" (designating the range as being above 1,000,000; or, more generally, as being extremely large).
The prefix "ump-" is added to the first two suffixes to produce the empty numbers "umpteen" and "umpty": it is of uncertain origin. There is a noticeable absence of an empty number in the hundreds range.
Usage of empty numbers:
- The word "umpteen" may be used as an adjective, as in "I had to go to umpteen stores to find shoes that fit." It can also be used to modify a larger number, usually "million", as in "Umpteen million people watched the show; but they still cancelled it."
- "Umpty" is not in common usage. It can appear in the form "umpty-one" (parallelling the usage in such numbers as "twenty-one"), as in "There are umpty-one ways to do it wrong." "Umpty-ump" is also heard, though "ump" is never used by itself.
- The word "zillion" may be used as an adjective, modifying a noun. The noun phrase normally contains the indefinite article "a", as in "There must be a zillion sites on the World Wide Web."
- The plural "zillions" designates a number indefinitely larger than "millions" or "billions". In this case, the construction is parallel to the one for "millions" or "billions", with the number used as a plural count noun, followed by a prepositional phrase with "of", as in "Out in the countryside, the night sky is filled with zillions of stars."
- Empty numbers are sometimes made up, with obvious meaning: "squillions" is obviously an empty, but very large, number; a "squintillionth" would be a very small number.
- Some empty numbers may be modified by actual numbers, such as "four zillion", and are used for jest, exaggeration, or to relate abstractly to actual numbers.
- Empty numbers are colloquial, and primarily used in oral speech or informal contexts. They are inappropriate in formal or scholarly usage.
See also Placeholder name.
See also
References
- ^ MROB
- ^ Yahoo!
- ^ "Hat trick, n.". Oxford English Dictionary. Oxford University Press. Retrieved 26 December 2014.
- ^ "Shock, n.2". Oxford English Dictionary. Oxford University Press. Retrieved 26 December 2014.
- ^ http://dictionary.reference.com/browse/ton?s=t
- ^ http://www.npr.org/templates/story/story.php?storyId=120470656
- ^ What is a partitive numeral?
- ^ Gary Blake and Robert W. Bly, The Elements of Technical Writing, pg. 22. New York: Macmillan Publishers, 1993. ISBN 0020130856
External links
- English Numbers - explanations, exercises and number generator (cardinal and ordinal numbers)