English Grammar
Table of contents
- Indefinite Articles “a” and “an”
- Plural
- The Definite Article “the”
- Countable and Uncountable Nouns
- Demonstrative pronouns "This", "That", "These" and "Those"
- The time
- Ordinal numbers
- The imperative
- The Saxon genitive
- Personal pronouns in English
- Possessive adjectives in English
- Possessive pronouns in English
- Reflexive Pronouns
- Practice with AI
- Introduction to Verb Tenses
- Simple Tenses
- Continuous Tenses
- Perfect Tenses
- The verb "to be"
- The verb "to have"
- Other verbs
- Glossary
- Author's Note
I recently decided to conclude my English language studies and obtain a certification, it should be noted that I have not studied the language for more than three years and my current knowledge is more focused on technicalities related to programming.
That said, this post will be a compilation of all English grammar, which I have begun to study in a self-taught way, also every week I will add 5 new topics which I will accompany with links to online exercises.
My goal is to create a complete compilation that allows readers to understand in a simple way something that many times in a classroom can be confusing.
Indefinite Articles “a” and “an”
Translation
| Indefinite Article | Translation |
|---|---|
| a | a/one |
| an | a/one |
When to use "a" or "an"?
The meaning is the same; the difference is that we will use "a" when the word begins with a consonant sound and "an" when the pronunciation begins with a vowel.
| Indefinite Article | Application | Examples |
|---|---|---|
| a | It is used before words that begin with a consonant sound. | a dog, a car, a university ("university" sounds "yu", starts with consonant sound "y") |
| an | It is used before words that begin with a vowel sound. | an apple, an elephant, an hour |
Watch out for the "h"
If the word begins with "h", the indefinite article to be used will vary depending on the pronunciation.
| Condition | Indefinite Article | Examples |
|---|---|---|
| "H" is pronounced | We will use "a" | a house, a hospital |
| "h" is not pronounced | We'll use "an" | an honest man, an hour, an honor |
Here are some words where the "h" is silent:
- honest
- Honorable
- hour
- Heir
- Herb
- Honor
When Not to Use the Indefinite Article
- Not used with countless
- Not used with plural nouns
Note: From here we can already see the importance of learning and practicing the pronunciation of words, something that will be constant in most subjects.
Exercises
🔗 The Indefinite Article : English exercise.
🔗 Articles – Open Access Exercise
🔗 Indefinite Articles: A or An? - English grammar exercises
Plural
There are a few rules we can use to quickly know the plural of most words:
| Condition | Ruler | Example |
|---|---|---|
| Words ending in "s, ss, x, ch, sh, z" | "Es" is added | class > classes |
| "Consonant" + "and" | The "y" becomes "i" and "is" is added | city > cities |
| Words that end in "f" or "faith" | The ending is removed and "ves" is added | knife > knives |
| Words that end in "o" | They are generally converted to the plural by adding "es" | potato > potatoes |
Irregular plurals
Likewise, there are irregular words, which do not follow the above rules, some examples are:
| Singular | Plural |
|---|---|
| mouse | MICE |
| scarf | scarves, scarfs |
| man | man |
| Woman | Women |
| person | people ("persons", in official documents) |
Exercises
🔗 Plurals – Exercise of free access
🔗 Noun Plural - English Grammar Exercises
The Definite Article “the”
Translation
the => the / the / the / the
When to Use “the”
| Use Case | Example |
|---|---|
| Countable nouns | The book on the table is mine |
| Refer to something already mentioned | I saw a dog. The dog was very friendly |
| Talking about something or someone at the moment we identify it | Close the door, please |
| Before the proper name of a sea, river, archipelago or mountain range | The Pacific Ocean, the Nile, the Bahamas, the Andes |
| Before a singular noun that designates a class of animals or things | The lion is a strong animal |
| When it accompanies non-countable nouns without talking about them generically, but referring to a specific object | The water in this bottle is cold |
| For family names, by adding an "s" to the end of the last name | The Spencers |
| Countries that are plural or start with an adjective | The Netherlands, The United States, The United Kingdom |
| In the cardinal points when the preposition "of" is used | The north of Mexico is very dry |
When Not to Use “the”
| NO Use Case | Example |
|---|---|
| Usually with non-countable nouns | Milk is a very common food |
| When using a non-countable noun in a generic mode | Water is essential for life |
| For academic subjects, since they are uncountable, therefore, they do not require the definite article | She studies Math and Biology |
| In front of country names in general | Spain, Mexico, Japan |
| With the cardinal points | Birds migrate south in winter |
| When a plural noun indicates an object generically | Cats are independent animals |
Pronunciation
| Condition | Pronunciation | Example |
|---|---|---|
| The + word that begins with consonant sound | /ðə/ (daa) | the car, the dog, the book |
| The + word that begins with vowel sound | /ðiː/ (dii) | the apple, the elephant, the hour |
| When you want to emphasize something | /ðiː/ (dii) | this is the (dii) best day ever! |
Exercises
🔗 The Definite Article : English Exercise.
🔗 The Definite Article 2 : English Exercise.
Countable and Uncountable Nouns
Within the English language we can differentiate two types of nouns, countable and non-countable:
- water: not countable, you can't say "one water", "two waters"...
- house: accountant, you can say, "one house", "two houses"...
Key Differences
| Accountants | Uncountable | Example |
|---|---|---|
| They have plural | They do not have plural | house > houses / water > *waters |
| They can go with the indefinite article (a/an) | They do not use the indefinite article (a/an) | a hause / *a water |
| - | They are always accompanied by the singular verb | water is good > water is good; "is" is the singular of the verb tobe |
Examples of Uncountable Nouns
- wine
- Water
- milk
- Beer
- cheese
- fruit
Turning Uncountable Nouns into Countable Nouns
Many non-countable nouns can sometimes become countable nouns, however, their meaning changes:
- "wine" (non-countable noun) => wine is red
- "wine" (countable noun) => Sherry is a wine from Spain
In the second case, wine indicates a type of wine, not wine in general
- "hair" (countable noun) => hair / hairs means "hair/hairs" and refers to one or more
- "hair" (non-countable noun) => refers to hair
Physical discomfort
Physical discomfort is compound words that end in "ache"
In Great Britain, the following are non-countable:
- I've got headache => I have a headache
In the United States, the following are considered accountants:
- I've got a headache => I have a headache
Demonstrative pronouns "This", "That", "These" and "Those"
| Pronoun | Meaning |
|---|---|
| This | This, this, this |
| That | That, that, that, that, that |
| These | These, these |
| Those | Those, those, those, those, those, thos |
That is a table > That's a table (more colloquial form)
Determiner
Demonstrative pronouns also function as determiners:
- Pronoun: substitutes for noun => My opinion is this
- Determiner: accompanies the noun => I'm tired of these jeans
Exercises
🔗 Demonstrative Pronouns : English exercise.
🔗 Exercises: Demonstrative pronouns
The time
Key Concepts
| On the dot | O'clock |
| Fourth | quarter |
| And a half | Half past |
| For minutes within the first half hour | Past |
| For minutes within the second half hour | to |
| Midnight | Midnight |
| Noon | noon |
| AM & PM | They are used the same as in Spanish |
Examples
| It's five past ten | It's ten past five |
| It's ten past eleven | It's ten past eleven |
| It's twenty-five to four | It's twenty-five minutes to four |
| It's quarter to seven | It's a quarter to seven |
| It's half past two | It's half past two |
| It's twelve o'clock | It's twelve o'clock |
| It's one to twelve | It's one minute to twelve |
| It's midnight | It's midnight |
| It's noon | It's noon |
Colloquial form
An easier and more colloquial way to tell time is like we would read it on a digital clock:
| Six twenty | The Six Twenty |
| Eight thirty | Half past eight |
| Ten fifty | Ten fifty |
Notes
- To ask the time we can say: "What time is it?"
- An easy way to remember Past and to It's with the word "pasture":
- Past → first half of the hour
- to → second half
Exercises
🔗 The Time – Free Access Exercise
🔗 Exercise 1 - The time in English
Ordinal numbers
Rules
| Add "th" to the cardinal number | four → fourth |
| Tens change the final "y" to "ie" before adding "th" | twenty → twentieth |
| When ordinals are written in ciphers, the last two letters of the written number are added | one hundred and forty-second → 142nd |
Exceptions
Exceptions are like irregular verbs, there is no rule or pattern, you just have to learn them.
| First | first |
| Second | second |
| Third | third |
| Fifth | fifth |
| Eighth | eighth |
| Ninth | ninth |
| Twelfth | twelfth |
Exercises
🔗 Ordinal Numbers : English exercise.
🔗 Ordinal numbers – Exercise of free access
The imperative
Generally, the Imperative has the same shape as the infinitive (to + verb), but without the "to".
Uses
| Give orders | Turn the lights on! → Turn on the lights! |
|---|---|
| Give instructions or directions | urn right at the traffic light. → Turn right at the traffic light. |
Negative form
It is formed with don't + verb.
Example:
Don't do it! → Don't do it!
Emphasize More
For reinforce the imperative, the verb must be preceded by "do", and serves to express Emphasis or irritation.
Example:
Do turn it off! → Turn it off!!
Use in instructions
When the imperative is used to give instructions, the verb can be in its form infinitive with "to", especially in educational or explanatory contexts.
Examples:
Listen to these words. → Listen to these words.
Try to pronounce this word. → Try to pronounce this word.
Exercises
🔗 Imperative Sentences : English exercise.
🔗 Imperative Sentences 2 : English exercise.
🔗 Imperative – Exercise of free access
The Saxon genitive
The Saxon genitive is used for denote possession in the following cases:
- People
- Pets
- Countries
- Expressions of time, space, or distance
- Sometimes, for cars and boats
It is also used with Collective Names that refer to Bodies, companies or institutions, especially when referring to the people who work in them.
Examples:
- The hospital's room → The hospital room
- The company's director → The director of the company
Structure
Possessor's Name + 's + Possessed Object Name
Examples:
- Susan's book → Susan's Book
- Tom's grandmother → Tom's Grandmother
Nouns ending in "s"
When the noun is plural and ends in "s", only the apostrophe (').
Example:
- Two minutes' walk → A two-minute walk
Use in Time Expressions
Example:
- Have you got today's paper? → Do you have today's newspaper?
Double Possessive
When there is Two possessives (Saxon genitive), the order of the words is Invest with respect to Spanish.
Examples:
- This is John's mother's car. → This is John's mother's car
- Take Emily's sister's book, please. → Take Emily's sister's book, please
Difference Between British and American English
The main difference lies in the use with Collective Names (restaurants, shops, schools, churches, etc.):
- British: requires the Saxon genitive with apostrophe and "s". → St. Mary's School
- American: Leave the name No changes, since possession is implied. → St. Mary School
Exercises
🔗 Exercise 1 - Genitive Saxon in English - 1st ESO - Online English Exercises
Personal pronouns in English
Personal pronouns Subject
They are used to indicate Who performs the action of the verb.
| Personal pronoun | Translation |
|---|---|
| I | I |
| You | You |
| He | He |
| She | She |
| It | It/It |
| We | About Us |
| You | You |
| They | They / Them |
Example:
- I play soccer. → I play soccer.
- They study English. → They study English.
Personal pronouns Object
They are used when the pronoun Get the action from the verb or it goes after a preposition.
| Personal pronoun object | Translation |
|---|---|
| I | Me |
| You | You |
| Him | He |
| Her | She |
| It | It |
| Us | About Us |
| You | You |
| Them | They / Them |
Example:
- I dance with her. → I dance with her.
- You talk to me. → You talk to me.
Important Note
Don't confuse the Subject pronouns with the object pronouns.
Although "you" and "it" are spelled the same in both cases, Its function changes according to the place they occupy in the sentence.
Example:
- You love me. → You love me (you is subject).
- I love you. → I love you (you is an object).
Exercises
🔗 Pronouns Staff : English exercise.
🔗 Exercises: Subject and object pronouns
🔗 Personal Pronouns – Exercise of Free Access
Possessive adjectives in English
The possessive adjectives In English they are words that indicate who owns something and are placed before the noun.
Examples:
My → me, my → My ticket = my ticket / My tickets = My Tickets
Your → you, your → Your house = your house / Your houses = your houses
Features
- The order is the same as in Spanish: possessive adjective + noun.
- They do not differentiate between male and female.
- They do not change between singular and plural.
- Unlike Spanish, they agree with the possessor and not with the thing possessed.
List of possessive adjectives
| My | my, my |
| Your | tu, tus (for "you") |
| His | su, sus (when the possessor is male) |
| Her | su, sus (when the possessor is female) |
| It's | his, his (for animals or objects) |
| Our | our, our, our, ours |
| Your | his, his (for "you") |
| Their | their, their (for "they" or "they") |
Use with human body parts
In English, to talk about parts of the body the adjective possessive is always used, and not the definite or indefinite article as in Spanish.
Example:
I've got a pain in my arm → My arm hurts.
(Not said I've got a pain in the arm.)
Exercises
🔗 Possessive adjectives and subject pronouns (I/my, you/your, etc.) - Test-English
Possessive pronouns in English
Example:
This is not my (possessive adjective) car. Is it yours (possessive pronoun)?
To understand possessive pronouns, it is very important to distinguish between a Pronoun and a adjective possessive.
In the previous sentence, My it is a possessive adjective because it accompanies the noun Car.
On the other hand, yours not only does it indicate possession, but replaces the noun (Car), since this is self-evident.
That is precisely a possessive pronoun.
Structure
The possessive pronouns (in English possessive pronouns) are formed, in most cases, Adding an "S" to the corresponding possessive adjective, as shown below:
| My | Mine | Mine, mine |
| Your | Yours | Yours, yours |
| His | His | His (his) |
| Her | Hers | Yours (hers) |
| Our | Ours | Ours, our |
| Your | Yours | Yours (yours) |
| Their | Theirs | Yours, yours (theirs) |
Exceptions
The exceptions are:
- My, which becomes mine.
- His, which does not add an extra "s".
- It's, which has no form of possessive pronoun, since it is not used to refer to the possession of things or animals without repeating the noun.
Exercises
🔗 Possessives: English exercise.
🔗 Possessive Pronouns and Determiners – Freely Accessible Exercise
🔗 Exercises - Possessive Adjectives and Possessive Pronouns
Reflexive Pronouns
The reflexive pronouns are used to indicate that the Subject and the object of a sentence are the same person or thing, or to Emphasize to the subject.
They are formed by adding "self" (for singular) or "Selves" (for plural) to the possessive adjectives.
List of Reflexive Pronouns
| My | Myself | Myself/Myself |
| Your | Yourself | Yourself/Yourself |
| His | Himself | himself/himself |
| Her | Herself | Herself |
| It's | Itself | Himself/Self |
| Our | Ourselves | ourselves |
| Your | Yourselves | yourselves |
| Their | Themselves | themselves |
Note:
- Himself and themselves are formed from the object pronouns him and them.
- Itself it only has an "s".
Characteristics of reflexive pronouns
Reflexive pronouns are usually accompanied by "by", and have the same meaning as expressions such as on my own (on my own, on my own) or on his own (on his own, by himself).
Example:
We went to the cinema by ourselves → We went to the movies on our own.
When we want reinforce the idea of doing it alone or without help, it is added "all":
Did you do your homework all by yourself? → Did you do your homework on your own?
Other uses
Reflexive pronouns are also used to underline or emphasize the subject of the sentence.
Examples:
I've been to England myself → I have been to England myself.
He likes to go running himself → He himself likes to go running.
Difference Between Reflexive and Reciprocal Phrases
In English, a distinction is made between the reflexive phrases and the reciprocal.
Examples:
- Reflexive: John and Mary hate themselves when they fight.
→ John and Mary hate themselves when they fight. - Reciprocal: John and Mary hate each other when they fight.
→ John and Mary hate each other when they fight.
Note:
Each other is used for Two people or things, while one another Used when there is more than two.
Example:
The four teammates hate one another → The four teammates hate each other.
Exercises
🔗 Reflexive Pronouns : English exercise.
🔗 Reflexive Pronouns 2 : English exercise.
🔗 Reflexive Pronouns – Free Access Exercise
🔗 Exercises: Reflexive pronouns
Practice with AI
From this point we will begin to study the Verb tenses, for which I recommend Combined exercises that include phrases where all the concepts seen above are applied.
The objective is to Strengthen the bases of your understanding of the English language.
I also recommend using the Artificial Intelligence with the following prompt:
"Make a list of 10 mixed multiple-choice exercises with sentences involving object pronouns, possessive adjectives, possessive pronouns, and reflexive pronouns."
Example:
Once you complete the exercises, ask for more and Repeat them until you manage to solve them correctly consecutively.
Remember that the AI is a tool, and as such, You can still make mistakes. Fear not questioning, questioning, or correcting: will not take it personally. Knowledge is free.
Introduction to Verb Tenses
In English, the Verb tenses they indicate When the action occurs (present, past or future) and How it develops (whether it is a simple, continuous or perfect action).
Here's an overview of the three main types:
Simple Tenses
They are made up of A single verb, and in some cases, a Auxiliary verb.
Example: I walk.
Continuous Tenses
They are formed with the verb "to be" (meaning "to be") plus the Main verb in gerund (-ing).
Example: I'm walking.
Perfect Tenses
They are formed with the verb "to have" (meaning "to have") plus the Main verb in participle (past participle).
Example: I've walked.
Auxiliary Verbs in Simple Tenses
When we talk about Auxiliary verbs in the Simple tenses, we are mainly referring to "do" and their conjugations "does" (present) and "did" (past).
These are used to forming negative and interrogative sentences.
Simple Tenses
The Simple tenses They are divided into present, past, future, and conditional, and each of them can be expressed in affirmative, negative and interrogative form.
Simple Present
- Affirmative: I'm walking
- Negative: I don't walk
- Interrogative: Am I walking?
Simple Past
- Affirmative: I was walking
- Negative: I wasn't walking
- Interrogative: Was I walking?
Simple Future
- Affirmative: I'll walk
- Negative: I won't walk
- Interrogative: Will I walk?
Simple Conditional
- Affirmative: I'd walk
- Negative: I wouldn't walk
- Interrogative: Would I walk?
Continuous Tenses
Continuous tenses are divided into present, past, future, and conditional, and each one has a affirmative, negative and interrogative.
Structure
Continuous tenses are formed with:
- The verb "to be" Conjugate
- The Main verb in gerund (completion -ing)
Present Continuous
- Affirmative: I'm walking
- Negative: I'm not walking
- Interrogative: Am I walking?
Past Continuous
- Affirmative: I was walking
- Negative: I wasn't walking
- Interrogative: Was I walking?
Future Continuous
- Affirmative: I'll be walking
- Negative: I won't be walking
- Interrogative: Am I walking?
Conditional Continuous
- Affirmative: I'd be walking
- Negative: I wouldn't be walking
- Interrogative: Was I walking?
Perfect Tenses
Perfect tenses are divided into present, past, future, and conditional, and each one has a form affirmative, negative and interrogative.
Structure
They are formed with:
- The verb to have conjugated
- The Main verb in participle
Present Perfect
- Affirmative: I've walked
- Negative: I haven't walked
- Interrogative: Have I walked?
Past Perfect
- Affirmative: I had walked
- Negative: I hadn't walked
- Interrogative: Had I walked?
Future Perfect
- Affirmative: I'll have walked
- Negative: I won't have walked
- Interrogative: Have I walked?
Conditional Perfect
- Affirmative: I would have walked
- Negative: I wouldn't have walked
- Interrogative: Would I have walked?
The verb "to be"
The verb to be is used to describe identity and states or conditions, including characteristics, professions, locations, and more.
Affirmative Form
| Meaning | Without contracting | Contracted |
|---|---|---|
| I am / I am | I am | I'm |
| You are / You are | You are | You're |
| He is/He is | He is | He's |
| She is / She is | She is | She's |
| It is / It is | It is | It's |
| We are / We are | We are | We're |
| You are / You are | You are | You're |
| They are / They are | They are | They're |
Negative formation
| Meaning | Without contracting | Contracted |
|---|---|---|
| I'm not / I'm not | I am not | I'm not |
| You're not / You're not | You are not | You aren't |
| He's not / He's not | He is not | He isn't |
| She's not / She's not | She is not | She isn't |
| It is not / It is not | It is not | It isn't |
| We are not / We are not | We are not | We aren't |
| You are not / You are not | You are not | You aren't |
| They are not / They are not | They are not | They aren't |
Interrogative Form
(It has no contraction)
| Meaning | Without contracting |
|---|---|
| Is it me? / Am I there? | Am I? |
| Is that you? / Are you here? | Are you? |
| Is it him? / Is he there? | Is he? |
| Is it her? / Is she here? | Is she? |
| Is that it? / Is it? | Is it? |
| Is it us? / Are we there? | Are we? |
| Is it you? / Are you here? | Are you? |
| Is it them? / Are they there? | Are they? |
Exercises
🔗 Exercises: 'To be' - Simple present
The verb "to have"
The verb to have can express possession and serve as an auxiliary verb. It is used to express possession and to form the perfect tenses in English.
Affirmative Form
| Meaning | Without contracting | Contracted |
|---|---|---|
| I have / I have | I have | I've |
| You have / You have | You have | You've |
| He has/He has | He has | He's |
| She has / She has | She has | She's |
| It has / It has | It has | It's |
| We have / We have | We have | We've |
| You have/You have | You have | You've |
| They have/They have | They have | They've |
Negative formation
| Meaning | Without contracting | Contracted |
|---|---|---|
| I don't have / I haven't | I have not | I haven't |
| You don't have / You haven't | You have not | You haven't |
| He doesn't have / He hasn't | He has not | He hasn't |
| She doesn't have / She hasn't | She has not | She hasn't |
| It doesn't have / It hasn't | It has not | It hasn't |
| We don't have / We haven't | We have not | We haven't |
| You don't have / You haven't | You have not | You haven't |
| They don't have / They haven't | They have not | They haven't |
Interrogative form
(It has no contraction)
| Meaning | Without contracting |
|---|---|
| Do I have? / Have I? | Have I? |
| Do you have? / Have you? | Have you? |
| Does he have? / Has he? | Has he? |
| Does she have? / Has she? | Has she? |
| Does it have? / Has it? | Has it? |
| Do we have? / Have we? | Have we? |
| Do you have? / Have you? | Have you? |
| Do they have? / Have they? | Have they? |
Important Note
The conjugation we have just seen is the one used for the Perfect tenses in English (present perfect, past perfect, future perfect, etc.).
However, the use of "to have" to express possession is explained in the next section.
Other verbs
For all verbs except modal verbs (can, could, must, have to, might, may, should), these rules apply.
We will use the verb to walk as an example.
Affirmative form
(It has no contraction)
| Meaning | Without contracting |
|---|---|
| I walk | I walk |
| You walk | You walk |
| He walks | He walks |
| She walks | She walks |
| It walks | It walks |
| We walk | We walk |
| You walk | You walk |
| They walk | They walk |
Negative form
| Meaning | Without contracting | Contracted |
|---|---|---|
| I don't walk | I do not walk | I don't walk |
| You don't walk | You do not walk | You don't walk |
| He doesn't walk | He does not walk | He doesn't walk |
| She doesn't walk | She does not walk | She doesn't walk |
| It does not walk | It does not walk | It doesn't walk |
| We don't walk | We do not walk | We don't walk |
| You don't walk | You do not walk | You don't walk |
| They don't walk | They do not walk | They don't walk |
Interrogative form
(It has no contraction)
| Meaning | Without contracting |
|---|---|
| Do I walk? | Do I walk? |
| Do you walk? | Do you walk? |
| Does he walk? | Does he walk? |
| Does she walk? | Does she walk? |
| Does it walk? | Does it walk? |
| Do we walk? | Do we walk? |
| Do you walk? | Do you walk? |
| Do they walk? | Do they walk? |
Exercises
🔗 Exercises of Simple present affirmative with 'To have'
🔗Verb "To have" - Exercises - English A1
Glossary
Consonant
They are alphabet sounds that are spoken by blocking the air with the tongue, lips, or teeth.
- Examples: B, C, D, F, G, H, J, K, L, M...
Vowel
These are sounds that are produced by letting the air out freely, without blocking it.
- In English (and Spanish) they are: A, E, I, O, U
Noun
It is a word that names a person, place, thing or idea.
- Examples: dog, teacher, city, love
Pronoun
It is a word that replaces a noun so as not to repeat it.
- Examples: he, she, it, we, they
Prompt
A prompt is an instruction, order, or input given to an AI tool to generate a response.
Author's Note
Writing this grammar compilation has been very rewarding for me, however, today for different reasons it is very difficult for me to keep track of 5 topics every week. This post will be paused since for now (June 2026) my time and attention are focused on other projects, however, I will still make gradual revisions of what has already been addressed and I will try to improve the way in which the information is organized and make it more inclusive through various tools such as the use of diagrams, tables and text-to-audio transcription.
Also, I hope to resume this post at the end of this year since at the end of it I intend to continue with a new one focused on algebra and calculus.
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