Cardinal numbers of a set

Cardinal Numbers - Meaning, Examples, Sets

Cardinal numbers are numbers that are used for counting. They are also known as natural numbers or cardinals. A set of cardinal numbers starts from 1 and it goes on up to infinity. We use cardinal numbers to answer the question "how many?". For example, how many students are going to the school picnic? The answer could be any number like 20, 23, 30, etc. So, all these numbers come in the category of cardinal numbers. In this article, we will explore the world of cardinal numbers and understand the difference between cardinal and ordinal numbers.

What are Cardinal Numbers?

A cardinal number describes or represents how many of something are present. Example 2 apples, 5 flowers, etc. It quantifies an object. It does not have values as fractions or decimals. Cardinal numbers are counting numbers, they help to count the number of items. Let's have a look at cardinal numbers examples. Ana wants to count the number of people standing in a queue at a billing counter. Can you help her? Ana started to count using Natural numbers.

Ana counted 1, 2, 3, 4, and 5. There are 5 people standing in a queue at the billing counter. Counting numbers are cardinal numbers! Now, Let's consider another example, Noah kept eight apples in a basket. The number eight denotes how many apples are there in the basket, irrespective of their order.

Examples of cardinal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and so on. The smallest cardinal number is 1 as 0 is not used for counting, so it is not a cardinal number.

Difference Between Cardinal and Ordinal Numbers

All the natural numbers are also referred to as cardinal numbers. Cardinal numbers are used for counting. While an ordinal number is a number that denotes the position or place of an object. Example: 1^st, 2^nd, 3^rd, 4^th, 5^th, etc. Ordinal numbers are used for ranking. Here is an example that explains cardinal and ordinal numbers:

In the above image, we can see a team of 4 workers on the construction site. This is an example of cardinal numbers.

In the above image, we can see the position of the runners in the running event. First, second, third, and so on. This is an example of ordinal numbers. Let's discuss ordinal and cardinal number differences in the table below:

Cardinal Numbers	Ordinal Numbers
They are counting numbers that represent quantity.	They are based on the rank or position of an object in a given list or order.
Cardinal numbers give us the answer of 'how many?'.	Ordinal numbers give us the answer of 'where'. For instance, where does the object lies in the list?
Examples are 1, 2, 3, 4, 5, 10, etc.	Examples are 1st, 2nd, 3rd, 4th, 5th, 10th, etc.

List of Cardinal Numbers from 1 to 100

Given below are the basic and most important cardinal numbers, which form the base for other counting numbers.

Given below is the list of all cardinal numbers from 1 to 100. It will also help you to see how we write cardinal numbers in words like 21- twenty-one.

Cardinal Numbers of a Set

In the case of a set, the cardinal number is the total number of elements present in it. In other words, the number of distinct elements present in a set is the cardinal number of the set. The cardinal number of a set A is represented as n(A). For example, the cardinal number of set W = {1, 3, 5, 7, 9} is n(W)=5, as there are 5 elements in it.

Think Tank:

• Maria wrote January 1, 2020, as today’s date in her notebook. Does the number 1 in the date, represent a cardinal number?
• There are _______ people in the line ahead of me. I am the 7th person in the line.

Important Notes:

• Cardinal numbers help us to count the number of things or people in or around a place or a group.
• The collection of all the ordinal numbers can be denoted by the cardinal.
• Cardinal numbers can be written as words such as one, two, three, etc.
• Cardinal numbers tell how many items, whereas ordinal numbers show position or ranking.

Related Articles

Check out these interesting articles to know more about cardinal numbers and its related articles.

• Ordinal Numbers
• Whole Numbers
• Natural Numbers
• Difference Between Natural and Whole Numbers

Cardinal Numbers Examples

1. Example 1: Kate has a list of numbers as shown - 7, 8^th, 10, Two, Fourth, 2^nd. Identify the cardinal numbers.

   Solution:

   7, 10 and two help us in counting, whereas 8^th, fourth and 2^nd helps us in identifying the position. Thus, 7, 10, and two are cardinal numbers.

2. Example 2: Help Ryan, to calculate the number of vowels in "NUMBERS". Also, identify the number of alphabets used to form this word.

   Solution:

   (i) We know that a, e, i, o, and u are the vowels and in the given word u and e are used. Therefore, 2 vowels are used to form this number (2 is the required cardinal number).

   (ii) We start counting from N to S, we see that "Numbers" has 7 alphabets in all. Therefore, a total of 7 alphabets are required to form the given word and here 7 is a cardinal number.

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Practice Questions on Cardinal Numbers

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FAQs on Cardinal Numbers

What is a Cardinal Number Example?

Cardinal numbers are used for counting. Some examples of cardinal numbers are 1, 2, 3, 4, 5, 10, 15, 20, 30, 40, 50, 100, etc. In our daily life, we use cardinal numbers a lot. Even a small child uses this mathematical concept without knowing the term for it. They do count how many toys they have, how many people are there around them, how many friends they have, how many subjects do they study at school, etc.

What is the Smallest Cardinal Number?

The smallest cardinal number is 1 (one) as whenever we count, we always start from 1.

How do you Find Cardinal Numbers?

Cardinal numbers can be found by counting. We start by 1 and then go on as per the number sequence.

How is Cardinal Number Different from the Ordinal Number?

Cardinal numbers are numbers that represent the number of items(quantity) while ordinal numbers represent the rank or position of an item in the given list.

What is a Cardinal Number in Sets?

In set A, if there are a total of 25 elements then 25 is the cardinal number of set A represented by n(A).

Can Cardinal Numbers Negative?

No, cardinal numbers cannot be negative. They are positive integers or natural numbers, as we always count the number of items starting from number 1, and then it goes up to infinity.

Is Zero a Cardinal Number?

No, 0 is not a cardinal number as cardinal numbers represent quantity, and 0 means nothing or no quantity.

What is the Biggest Cardinal Number?

There are infinite natural numbers. Therefore, there are as many cardinal numbers as natural numbers. There can be no generalization of the biggest natural number and so does for the biggest cardinal number.

cardinal number in nLab

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Context

Arithmetic

number theory

• arithmetic

• arithmetic geometry, arithmetic topology

• higher arithmetic geometry, E-∞ arithmetic geometry

number

• natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number

arithmetic

• Peano arithmetic, second-order arithmetic

• transfinite arithmetic, cardinal arithmetic, ordinal arithmetic

• prime field, p-adic integer, p-adic rational number, p-adic complex number

arithmetic geometry, function field analogy

• arithmetic scheme

  • arithmetic curve, elliptic curve

  • arithmetic genus

• arithmetic Chern-Simons theory

• arithmetic Chow group

• Weil-étale topology for arithmetic schemes

• absolute cohomology

• Weil conjecture on Tamagawa numbers

• Borger's absolute geometry

• Iwasawa-Tate theory

  • arithmetic jet space

• adelic integration

• shtuka

• Frobenioid

Arakelov geometry

• arithmetic Riemann-Roch theorem

• differential algebraic K-theory

Type theory

natural deduction metalanguage, practical foundations

• judgement

• hypothetical judgement, sequent

  • antecedents⊢\vdashconsequent, succedents

1. type formation rule
2. term introduction rule
3. term elimination rule
4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

• calculus of constructions

syntax object language

• theory, axiom

• proposition/type (propositions as types)

• definition/proof/program (proofs as programs)

• theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

homotopy levels

• type theory

• 2-type theory, 2-categorical logic

• homotopy type theory, homotopy type theory - contents

  • homotopy type

  • univalence, function extensionality, internal logic of an (∞,1)-topos

  • cohesive homotopy type theory

  • directed homotopy type theory

  • HoTT methods for homotopy theorists

semantics

• internal logic, categorical semantics

  • display map

• internal logic of a topos

  • Mitchell-Benabou language

  • Kripke-Joyal semantics

• internal logic of an (∞,1)-topos

  • type-theoretic model category

Edit this sidebar

• Idea
• Definition
• In homotopy type theory
• Cardinal arithmetic
• Definition
• Properties
• Properties of cardinals
• Related concepts
• References

Idea

The cardinal numbers (or just cardinals) constitute a generalisation of a natural numbers to numbers of possibly infinite magnitudes. Specifically, cardinal numbers generalise the concept of ‘the number of …’. In particular, the number of natural numbers is the first infinite cardinal number.

Definition

Naïvely, a cardinal number should be an isomorphism class of sets, and the cardinality of a set SS would be its isomorphism class. That is:

1. every set has a unique cardinal number as its cardinality;
2. every cardinal number is the cardinality of some set;
3. two sets have the same cardinality if and only if they are isomorphic as sets.

Then a finite cardinal is the cardinality of a finite set, while an infinite cardinal or transfinite cardinal is the cardinality of an infinite set. (If you interpret both terms in the strictest sense, then there may be cardinals that are neither finite nor infinite, without some form of the axiom of choice).

Taking this definition literally in material set theory, each cardinal is then a proper class (so one could not make further sets using them as elements). For this reason, in axiomatic set theory one usually defines a cardinal number as a particular representative of this equivalence class. There are several ways to do this:

• The cardinality of a set SS is the smallest possible ordinal rank of any well-order on SS. In other words, it is the smallest ordinal number (usually defined following von Neumann) which can be put in bijection with SS. A cardinal number is then any cardinality, i.e. any ordinal which is not in bijection with any smaller ordinal.

  • On well-orderable sets, this cardinality function satisfies (1–3), but one needs the axiom of choice (precisely, the well-ordering theorem) to prove that every set is well-orderable. This approach is probably the most common one in the presence of the axiom of choice.

  • In the absence of excluded middle, when the “correct” definition of well-order is different from the usual one (and so “the least ordinal such that …” may not exist), a better definition of the cardinality of SS is as the set of all ordinal numbers less than the ordinal rank of every well-order on SS.

• Alternatively, we can define the cardinality of a set XX to be the set of all well-founded pure sets that are isomorphic as sets to XX and such that no pure set of smaller hereditary rank (that is, which occurs earlier in the von Neumann hierarchy) is isomorphic to XX.

  • On those sets that are isomorphic to some well-founded set, this cardinality function satisfies (1–3), but one needs some assumption to prove that every set is isomorphic to a well-founded set. (This will follow directly from the axiom of foundation; it will also follow from the axiom of choice, since then every set is isomorphic to a von Neumann ordinal.)

In the absence of the appropriate axioms, the definitions above can still be used to define well-ordered cardinals and well-founded cardinals, respectively.

From the perspective of structural set theory, it is evil to care about distinctions between isomorphic objects, and unnecessary to insist on a canonical choice of representatives for isomorphism classes. Therefore, from this point of view it is natural to simply say:

• A cardinal is a set (that is, an object of Set).

However, one still may need sets of cardinals, that is sets that serve as the target of a cardinality function satisfying (1–3) on any (small) collection of sets. One can construct this as a quotient set of that collection.

Lowercase Greek letters starting from κ\kappa are often used for cardinal numbers.

In homotopy type theory

A cardinal in homotopy type theory is an element of the type of cardinals κ:Card𝒰\kappa \colon Card_\mathcal{U} relative to a universe 𝒰\mathcal{U}. The type of cardinals is defined as the set-truncation of the type of sets Set𝒰Set_\mathcal{U} relative to 𝒰\mathcal{U}, Card𝒰≔|Set𝒰|0Card_\mathcal{U} \coloneqq \left| Set_\mathcal{U} \right|_0 with Set𝒰Set_\mathcal{U} defined as

Set𝒰≔∑A:𝒰∏a:A∏b:A∏p:a=Ab∏q:a=Abp=a=AbqSet_\mathcal{U} \coloneqq \sum_{A:\mathcal{U}} \prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \prod_{q:a =_A b} p =_{a =_A b} q

Cardinal arithmetic

Definition

For SS a set, write |S|{|S|} for its cardinality. Then the standard operations in the category Set induce arithmetic operations on cardinal numbers (“cardinal arithmetic”):

For S1S_1 and S2S_2 two sets, the sum of their cardinalities is the cardinality of their disjoint union, the coproduct in SetSet:

|S1|+|S2|≔|S1⨿S2|. {|S_1|} + {|S_2|} \coloneqq {|S_1 \amalg S_2|} \,.

More generally, given any family (Si)i:I(S_i)_{i: I} of sets indexed by a set II, the sum of their cardinalities is the cardinality of their disjoint union:

∑i:I|Si|≔|∐i:ISi|. \sum_{i: I} {|S_i|} \coloneqq {|\coprod_{i: I} S_i|} \,.

Likewise, the product of their cardinalities is the cardinality of their cartesian product, the product in SetSet:

|S1||S2|≔|S1×S2|. {|S_1|} \, {|S_2|} \coloneqq {|S_1 \times S_2|} \,.

More generally again, given any family (Si)i:I(S_i)_{i: I} of sets indexed by a set II, the product of their cardinalities is the cardinality of their cartesian product:

∏i:I|Si|≔|∏i:ISi|. {|S|}, where Ω\Omega is the cardinality of the set of truth values.

The usual way to define an ordering on cardinal numbers is that |S1|≤|S2|{|S_1|} \leq {|S_2|} if there exists an injection from S1S_1 to S2S_2:

(|S1|≤|S2|):⇔(∃(S1↪S2)). ({|S_1|} \leq {|S_2|}) \;:\Leftrightarrow\; (\exists (S_1 \hookrightarrow S_2)) \,.

Classically, this is almost equivalent to the existence of a surjection S2→S1S_2 \to S_1, except when S1S_1 is empty. Even restricting to inhabited sets, these are not equivalent conditions in constructive mathematics, so one may instead define that |S1|≤|S2|{|S_1|} \leq {|S_2|} if there exists a subset XX of S2S_2 and a surjection X→S1X \to S_1. Another alternative is to require that S1S_1 (or XX) be a decidable subset of S2S_2. All of these definitions are equivalent using excluded middle.

This order relation is antisymmetric (and therefore a partial order) by the Cantor–Schroeder–Bernstein theorem (proved by Cantor using the well-ordering theorem, then proved by Schroeder and Bernstein without it). +}) is called Cantor’s continuum problem; the assertion that this is the case is called the (generalized) continuum hypothesis. It is known that the continuum hypothesis is undecidable in ZFC.

Properties of cardinals

• A transfinite cardinal π\pi is a regular cardinal if no set of cardinality π\pi is the union of fewer than π\pi sets of cardinality less than π\pi. Equivalently, π\pi is regular if given a function P→XP \to X (regarded as a family {Px}x∈X\{P_x\}_{x\in X}) such that |X|<π{|X|} \lt \pi and |Px|<π{|P_x|} \lt \pi for all x∈Xx \in X, then |P|<π{|P|} \lt \pi. Again equivalently, π\pi is regular if the category Set<π\Set_{\lt\pi} of sets of cardinality <π\lt\pi has all colimits of size <π\lt\pi. The successor of any infinite cardinal, such as ℵ1\aleph_1, is a regular cardinal. \lambda is the cardinality of the power set P(λ)P(\lambda), a cardinal π\pi is a strong limit iff the category Set<π\Set_{\lt\pi} is an elementary topos.

• An inaccessible cardinal is any (usually uncountable) regular strong limit cardinal. A weakly inaccessible cardinal is a regular limit cardinal.

• A cardinal κ\kappa is a measurable cardinal if some (hence any) set of cardinality κ\kappa admits a two-valued measure which is κ\kappa-additive, or equivalently an ultrafilter which is κ\kappa-complete.

• groupoid cardinality

• counting measure

• ordinal number

References

The original article is

• Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46 (1895) pp.481-512, reprinted from p. 282 on in Ernst Zermelo (ed.), Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Springer Berlin 1932 (online English translation)

The book

• Peter J Cameron, Sets, Logic and Categories (ISBN: 1-85233-056-2 )

contains a very readable account of ZFC and the definitions of both ordinal and cardinal numbers.

Any serious reference on set theory should cover cardinal numbers. The long-established respected tome is

• Thomas Jech Set Theory;

there are also some references listed at

• MathWorld and

• the English Wikipedia.

A good introduction to infinite cardinals is:

• Frank R. Drake, Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier, 1974.

For a much deeper treatment, which assumes most of the material in Drake as a prerequisite, see:

• Akihiro Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginning, Springer, 2003.

This is a very readable and freely available historical introduction:

• Akihiro Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in Higher Set Theory, Springer Lecture Notes in Mathematics 669, 1978. (pdf)

Standard approaches start with a material set theory, such as ZFC, whereas the approach here uses structural set theory as emphasized above. Since cardinality is isomorphism-invariant, it is easy to interpret the standard material structurally, although the basic definitions will be different. There does not seem to be a standard account of cardinality from within structural set theory.

For cardinal numbers in homotopy type theory, see chapter 10 of

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

For a critical discussion of the history of the meaning of Cantor’s “Kardinalen”, see

• William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen",

which argues that Cantor’s original meaning of set was more like what today is cohesive set and that his Kardinalen refer to the underlying set (see at flat modality).

Last revised on September 30, 2022 at 12:52:43. See the history of this page for a list of all contributions to it.

Natural numbers: definition, examples, properties

Definition of a natural number

Natural numbers are numbers that we use to count something specific, tangible.

These are what are called natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, etc.

ascending order. The first hundred can be viewed in the table.

What operations are possible on natural numbers

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Decimal notation of a natural number

At school, we go through the topic of natural numbers in the 5th grade, but in fact, a lot of things can be intuitively clear to us even earlier. Let's talk about important rules.

We regularly use numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. When writing any natural number, you can use only these digits without any other symbols. We write the numbers one by one in a line from left to right, using the same height.

Examples of correct writing of natural numbers: 208, 567, 24, 1467, 899112. These examples show us that the sequence of numbers can be different and some can even be repeated.

077, 0, 004, 0931 are examples of incorrect notation of natural numbers, because zero is on the left. The number cannot start from zero. This is the decimal representation of a natural number.

Quantitative meaning of natural numbers

Natural numbers carry a quantitative meaning, that is, they act as a tool for numbering.

Imagine that we have a banana 🍌 in front of us. We can record that we see 1 banana. In this case, the natural number 1 is read as "one" or "one".

But the term "unit" has another meaning: that which can be considered as a whole. An element of a set can be denoted by a unit. For example, any tree from a set of trees is a unit, any leaf from a set of leaves is a unit.

Imagine that we have 2 bananas 🍌🍌 in front of us. The natural number 2 is read as "two". Further, by analogy:

The main function of a natural number is to indicate the number of objects.

If the entry of a number matches the number 0, then it is called "zero". Recall that zero is not a natural number, but it can mean the absence. Zero items means none.

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One-digit, two-digit and three-digit natural numbers

One-digit natural number is such a number, which consists of one character, one digit. Nine single-digit natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.

Two-digit natural numbers are those that include two signs, two digits. The numbers may be repeated or different. For example: 88, 53, 70.

If the set of objects consists of nine and one more, then we are talking about 1 ten (“one dozen”) objects. If one ten and one more, then we have 2 dozen (“two tens”) and so on.

Basically, a two-digit number is a set of single-digit numbers, where one is written on the right and the other on the left. The number on the left shows the number of tens in the natural number, and the number on the right shows the number of units. There are 90 two-digit natural numbers in total.

Three-digit natural numbers - numbers, which include three characters, three digits. For example: 666, 389, 702.

One hundred is a set of ten tens. One hundred and another hundred - 2 hundreds. Let's add another hundred - 3 hundreds.

This is how a three-digit number is written: natural numbers are written one after the other from left to right.

The rightmost single digit indicates the number of units, the next one indicates the number of tens, the leftmost one indicates the number of hundreds. The number 0 indicates the absence of units or tens. So 506 is 5 hundreds, 0 tens and 6 ones.

Four-digit, five-digit, six-digit and other natural numbers are defined in the same way.

Multi-digit natural numbers

Multi-digit natural numbers consist of two or more characters.

1,000 is a set with ten hundred, 1,000,000 is a thousand thousand, and one billion is a thousand million. A thousand million, just imagine! That is, we can consider any multi-valued natural number as a set of single-valued natural numbers.

For example, 2873206 contains: 6 ones, 0 tens, 2 hundreds, 3 thousand, 7 tens of thousands, 8 hundreds of thousands and 2 millions.

How many natural numbers are there?

One-digit 9, two-digit 90, three-digit 900, etc.

Properties of natural numbers

We already know about the features of natural numbers. And now we will tell in detail about their properties:

Digits of a natural number and the value of the digit

Recall that its value depends on the position on which the digit stands in the record of the number. So, for example, 1123 contains: 3 units, 2 tens, 1 hundred, 1 thousand. At the same time, we can formulate it differently and say that in a given number 1123, the digit 3 is located in the units digit, 2 in the tens digit, 1 in the hundreds digit and 1 serves as the value of the thousands digit.

The digit is the position, the location of the digit in the record of a natural number.

Each digit has its own name. The most significant digits are always on the left, and the least significant ones are on the right. To remember faster, you can use a table.

The number of digits always corresponds to the number of digits in the number. This table contains the names of all digits for a number that consists of 15 characters. The following digits also have names, but they are used extremely rarely.

The lowest (lowest) digit of a multi-digit natural number is the units digit.

The highest (highest) digit of a multi-digit natural number is the digit corresponding to the leftmost digit in the given number.

You must have noticed that textbooks often put small spaces when writing multi-digit numbers. This is done so that natural numbers are easy to read. And also to visually separate different classes of numbers.

Class is a group of digits that contains three digits: units, tens and hundreds.

Decimal

People at different times used different methods of writing numbers. And each number system has its own rules and features.

Decimal number system is the most common number system in which ten digits are used to write numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

In the decimal system, the value of one and of the same digit depends on its position in the number entry. For example, the number 555 consists of three identical digits. In this number, the first digit from the left means five hundred, the second - five tens, and the third - five units. Since the value of a digit depends on its position, the decimal number system is called positional.

Question for self-test

How many natural numbers can be marked on the coordinate ray between points with coordinates:

1. 0 and 15;

2. 20 and 50;

3. 100 and 130?

Numerals

The study of English, as a rule, begins with an acquaintance with the letter and sound structure of the language. Also one of the first and important topics for mastering is the topic "Numerals".

Students should be able not only to tell some general information about themselves, for example, what is their name, where they come from, where they live, and so on, but they also need to know English numbers in order to learn how to talk about age, that is, how old, when they were born, say the date, month and year. In addition, numerals are used to designate house numbers, chapters and pages in books, magazines, telephone numbers, and for mathematical operations. This article details the cases of using English digits.

First of all, it is worth explaining what a numeral is in English. Numeral is a part of speech that denotes the number or order of objects, persons. Numerals are divided into two categories: quantitative (Cardinal Numerals) and ordinal (Ordinal Numerals).

Cardinal numbers indicate the number of objects or beings and answer the question "How many?" - "How?".

The table contains numbers from 0-20, as well as tens and compound numbers.

It should be noted that the ending -s is not added to the numerals hundred, thousand, million;

one hundred - seven hundred - 100 - 700
one thousand - six thousand - 1000 - 6000
one million – five million – 1,000,000 - 5,000,000

As mentioned above, cardinal numbers in English are used to denote house numbers, pages and chapters in books, bus numbers, and so on. In this case, the noun will have no article, and the numeral will come after the noun to which it refers:

Chapter three is dedicated to the description of the family. – Chapter three is devoted to the description of the family.

- Children! Open page one hundred and thirty-five and do exercise four. - Children! Turn to page 135 and do exercise 4.

Ordinal numbers indicate the order of objects, their serial number. These numerals are used in response to the question "Which?" - "Which the?".

Ordinal numbers are formed from cardinal numbers with the suffix –th. There are several rules for adding the -th suffix:

1. For compound numerals, the suffix -th is added to the last word: fifty-four (fifty-four) - the fifty-fourth - fifty-fourth.

2. The definite article the should be used before ordinal numbers: the tenth - the tenth, the fourteenth - the fourteenth, the thirtieth - the thirtieth.

3. Tens ending in -y change the suffix to -ie- + -th: twenty - the twentieth.

4. As with many rules, there are exceptions: the first is the first, the second is the second, the third is the third, the fifth is the fifth, the ninth is the ninth, the twelfth is the twelfth.

So, we have considered two categories of numerals and their formation. Now it is necessary to study in more detail the cases when and how one should use cardinal and ordinal numbers in English.

1. Telephone numbers. In English, telephone numbers differ from Russian not only in the number of digits, but also in the rules of reading. If in Russian we can read phone numbers in tens and hundreds, then the numbers in English are read in a completely different way, namely, we must name each number separately.

   For example:
   Please, put down my phone number: six, nine, three, oh, seven, five, two, eight, six. – Please write down my number: 6-9-3-0-7-5 2-8-6.

   Please note that in telephone numbers it is customary to pronounce zero as oh, not zero. Also in football, zero on the scoreboard is nil, and in tennis it is love.

   Manchester United wins the match with the score of three nil – Manchester United won the match 3-0.

2. Fractions. In English, simple fractions are called Fractions, and decimal fractions are called Decimal Fractions.

   In simple fractions, the numerator is expressed as a cardinal number, and the denominator is expressed as ordinal. An ordinal number takes the ending -s if the numerator is greater than one.

   1/2 - a/one half
   1/3 - a/one third
   1/4 - one fourth / a quarter
   1/5 - a/one fifth
   1/10 - a/one tenth
   1/25 - a /one twenty-fifth
   1/100 - a /one hundredth
   1/1225 - a / one thousand two hundred and twenty-fifth
   2/3 - two thirds
   3/4 - three quarters/quarters
   4/7 - four sevenths
   7/18 - seven eighteenths
   9/10 - nine tenths
   2 1/2 - two and a half
   3 1/4 - three and a quarter/a fourth
   2/5 ton - two fifths of a ton
   1/4 kilometer - a quarter of a kilometer
   1/2 kilometer - a half a kilometer

   As for decimal fractions, it should be remembered that in English, unlike Russian, a point is put instead of a comma:

   0. 3 - (zero) point three
   .4 - point four
   0.5 - (zero) point five
   4.8 – four point eight
   3.156 - three point one five six
   74.39 - seventy-four point three nine

3. Mathematical operations with English numbers. To denote the sign "=" in English, the verbs "to be" and "to equal" are used in the meaning of "equal to". And, as a rule, these verbs in this case are used in the singular. Consider below the basic mathematical operations:

   plus [plʌs] - plus
   minus [ˈmaɪnəs] - minus
   multiplied by [ˈmʌltɪplaɪd] - multiplied by
   divided by [dɪˈvaɪdɪd] - divided by
   equal [ˈiːkwəl] - equal to

   24 + 17 = 42 - twenty-four plus seventeen is forty-two
   67 - 13 = 54 - sixty-seven minus thirteen equals fifty-four
   45 x 3 = 135 - forty-five multiplied by three is one hundred and thirty-five
   45 : 9 = 5 - forty-five divided by nine equals five

   Multiplication can also be expressed using the word "times": 5 x 3 = 15 - Five times three is fifteen.

4. Dates. Dates in English can be expressed in years, day and month, or all together.

   Years in English are read as follows:

   If there is a need to clarify when an action took place - before our era or after our era, then the designations B.C. (Before Christ) and A.D. (Anno Domini):

   This event took place in 658 B.C. This event took place in 658 BC.
   What happened in 1645 A.D.? What happened in 1645 AD?

   As for specifying dates with days and months, there are several options for writing them:

   25th December
   25 December
   The twenty-fifth of December

   December 25
   December 25th
   December the twenty-fifth

5. Also numbers are used when we talk about percentages. In English, the word "percent" is denoted by the word "percent" and is always used after numerals in the singular:

   5% - five percent
   60% - sixty percent
   78% - seventy-eight percent
   194% - one hundred and ninety-four percent
   56. 37% - fifty-six point thirty-seven percent

6. We also use numbers and names of currencies to denote money in English. As a rule, the numeral comes first, then the name of the currency itself. If the expression contains a mention of kopecks, cents, then this part will be pronounced like an ordinary number. The name of the currency is used in the plural, so you must add 9 to its name0313 -s ending:

   $15–fifteen dollars
   €25,99 - twenty-five euros ninety nine (cents, euro cents)
   £2386 – two thousand three hundred and eighty six pounds
   ₴54,394– fifty-four thousand three hundred and ninety-four hryvnias
   ₱651.28 – six hundred and fifty-one roubles and twenty-eight kopecks
   ¥73,829 – seventy-three thousand eight hundred and twenty-nine yen
   (yen is an exception, the -s ending is not added).

   It is worth noting that in English the currency symbol on the letter comes before the number, but is read after: $15 - fifteen dollars.

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