Oxford Dictionaries
The Oxford Word of the Year 2025 is rage bait
Collins
The Collins Word of the Year 2025 is vibe coding
The Cambridge Dictionary
The Cambridge Dictionary Word of the Year 2025 is parasocial
Proverbs
If a job’s worth doing, it’s worth doing well.
No sweet without sweat.
No pains, no gains.
He that is once born, once must die.
No man knows when he shall die, although he knows he must die.
A dead bee makes no honey.
A dead mouse feels no cold.
Dead dogs bark not.
Praise a fair day at night.
Men fear death as children fear to go in the dark.
We must live by the living, not by the dead.
The devil is not black as he is painted.
Sooner begun, sooner done.
No time like the present.
One good turn deserves another.
Good hand, good hire.
Like fault, like punishment.
Such answer as man gives, such will he get.
Every man after his fashion.
Every one to his taste.
So many men, so many options.
All roads lead to Rome.
Diligence makes an expert workman.
Business is the salt of life.
No bees, no honey; no work, no money.
First come, first served.
Early to bed and early to rise, makes a man healthy, wealthy and wise.
Bread is the staff of life.
The belly carries the legs.
Eat to live and not live to eat.
After dinner sit awhile, after supper walk a mile.
An apple a day keeps the doctor away.
There is no royal road to learning.
Never too late to learn.
Everything must have a beginning.
Everything has an end.
Garbage in, garbage out.
Practise what you preach.
Example is better than precept.
Practice makes perfect.
Keep your mouth shut and your eyes open.
Let bygones be bygones.
A friend in need is a friend indeed.
God helps them that help themselves.
A good heart conquers ill fortune.
All happiness is in the mind.
Laughter is the best medicine.
A merry heart goes all the way.
Great hopes make great men.
Hope for the best.
Tomorrow is another day.
Every man has his faults.
To err is human.
Forgive and forget.
Every path has a puddle.
Life is short and time is swift.
Life begins at forty.
It is easy to be wise after the event.
Better be safe than sorry.
Everyman for himself, and God for us all.
Skill is no burden.
A little body often harbours a great soul.
Many small maka a great.
Time and tide wait for no man.
Now is now, and then was then.
Travel broadens the mind.
The language of truth is simple.
It is too late to call back yesterday.
Time flies.
Time will tell.
Times change and we with them.
My English Phrases List - December - 2025
be supposed to
You are supposed to listen to your parents.
slip/escape someone’s mind
His name slips/escapes my mind at the moment.
pay off
I finally paid off the loan.
beat the rush
I suggested that we get there by 11:30 to beat the lunchtime rush.
miss the boat
missed the boat on buying stock in the company early on
kettle of fish
I’m used to paddling, but those rapids are a whole other kettle of fish.
My English Words List - December - 2025
snowsuit
noun
arrogance
noun
Programming is the best antidote to arrogance I’ve come across — I make so many errors that I am continually reminded of my own fallibility.
gut
noun
The guts uphold the heart, and not the heart the guts.
ballpoint pen
noun
curiosity
noun
Curiosity killed the cat.
Curiosity is endless, restless, and useless.
chip
noun
The best carpenter makes the fewest chips.
wood chips were spread over the ground between the plants
caravan
noun
bought a caravan and drove cross-country to California
toadstool
noun
- a fungus that has an umbrella-shaped cap, one that is poisonous or unfit for food
A toadstool generally refers to a poisonous mushroom.
schoolchild
noun
Almost every schoolchild in America learns Lincoln was born in a Kentucky log cabin. — Frank Witsil, Detroit Free Press, 13 July 2023
soggy
adjective
Be sure to keep the soil moist but not soggy. — Sheryl Geerts, Better Homes & Gardens, 15 Dec. 2025
pal
noun
We’ve been pals since we were kids.
Catchphrases
master of your domain
where’s the beef?
every home should have one!
if anything can go wrong, it will
never mind
don’t worry, be happy
after all, tomorrow is another day
give me a hug
It’s good to talk
Michael Artin's Algebra Exercises
1 Matrices
Section 1 The Basic Operations
1.1. What are the entries \( a_{21} \), and \( a_{23} \) of the matrix A = \( \begin{bmatrix}
1 &2 &5 \\
2 &7 &8 \\
0 &9 &4
\end{bmatrix} \) ?
\( a_{21} = 2 \), and \( a_{23} = 8 \)
1.2. Determine the products AB and BA for the following values of A and B:
\( A = \begin{bmatrix}
1 &2 &3 \\
3 &3 &1
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
-8 &-4 \\
9 &5 \\
-3 &-2
\end{bmatrix} \)
\( AB = \begin{bmatrix}
1 &0 \\
0 &1
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
-20 &-28 &-28 \\
24 &33 &32 \\
-9 &-12 &-11
\end{bmatrix} \)
\(
A = \begin{bmatrix}
1 &4 \\
1 &2
\end{bmatrix}, \hspace{2mm}
B = \begin{bmatrix}
6 &-4 \\
3 &2
\end{bmatrix}
\)
\( AB = \begin{bmatrix}
18 &4 \\
12 &0
\end{bmatrix}, \hspace{4mm}
BA = \begin{bmatrix}
2 &16 \\
5 &16
\end{bmatrix} \)
1.4. Verify the associative law for the matrix product
\( ABC = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
0 &1 &2 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix} \)
Note: This is a self-checkingproblem.It won’t come out unless you multiply correctly. If you need to practice matrix multiplication, use this problem as a model.
\( (AB)C = \begin{bmatrix}
2 &3 &8 \\
1 &1 &3
\end{bmatrix}
\begin{bmatrix}
1 \\
4 \\
3
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)
\( A(BC) = \begin{bmatrix}
1 &2 \\
0 &1
\end{bmatrix}
\begin{bmatrix}
10 \\
14
\end{bmatrix}
= \begin{bmatrix}
38 \\
14
\end{bmatrix}
= 2 \begin{bmatrix}
19 \\
7
\end{bmatrix}
\)
Proverbs
Far from eye, far from heart.
Friends agree best at a distance.
Out of sight, out of mind.
Long absent, soon forgotten.
Nothing seek, nothing find.
Anger punishes itself.
A hungry man is an angry man.
Fear of death is worse than death itself.
He laughs best who laughs last.
The unexpected always happens.
Never judge from apperances.
Better to ask the way than go astray.
Like question, like answer.
It is not every question that deserves an answer.
Every beginning is hard.
Such beginning, such end.
Nurture is above nature.
Nurture and good manners maketh man.
Manners and money make a gentleman.
‘After you’ is good manners.
Curiosity is ill manners in another house.
Better the devil you know than the devil you don’t know.
There is nothing permannent except change.
There is nothing new under the sun.
The apple never falls far from the tree.
Boys will be men.
You pays your money and you takes your choice.
The customer is always right.
Content is happiness.
A little bird is content with a little nest.
刻几何原本序
唐、虞之世，自羲、和治曆，暨司空、后稷、工、虞、典乐五官者，非度数不为功。《周官》六艺，数与居一焉；而五艺者，不以度数从事，亦不得工也。襄、旷之于音，般、墨之于械，岂有他谬巧哉？精于用法尔已。故尝谓三代而上，为此业者盛，有元元本本、师傅曹习之学，而毕丧于祖龙之焰。汉以来，多任意揣摩，如盲人射的，虚发无效；或依拟形似，如持萤烛象，得首失尾。至于今而此道尽废，有不得不废者矣。
《几何原本》者，度数之宗，所以穷方圆平直之情，尽规矩准绳之用也。利先生从少年时，论道之暇，留意艺学，且此业在彼中所谓师傅曹习者，其师丁氏，又绝代名家也，以故极精其说。而与不佞游久，讲谈余晷，时时及之。因请其象数诸书，更以华文。独谓此书未译，则他书俱不可得论，遂共翻其要约六卷，既卒业而复之，由显入微，从疑得信。盖不用为用，众用所基，真可谓万象之形囿，百家之学海。虽实未竟，然以当他书，既可得而论矣。
私心自谓，不意古学废绝二千年后，顿获补缀唐、虞、三代之阙典遗义，其裨益当世，定复不小。因偕二三同志，刻而传之。
先生曰：「是书也，以当百家之用，庶几有羲、和、般、墨其人乎，犹其小者，有大用于此，将以习人之灵才，令细而确也。」余以谓小用大用，实在其人。如邓林伐材，栋梁榱桷，恣所取之耳。顾惟先生之学，略有三种，大者修身事天，小者格物穷理。物理之一端，别为象数，一一皆精实典要，洞无可疑，其分解擘析，亦能使人无疑。而余乃亟传其小者，趋欲先其易使人绎其文，想见其意理，而知先生之学可信不疑。大概如是，则是书之为用更大矣。他所说几何诸家，藉此为用，略具其自叙中，不备论。
吴淞徐光启书。
My English Phrases List - November - 2025
get a move on
You’d better get a move on, darling. If you don’t hurry you’ll miss the train.
Could you get a move on?
She told her partner to get a move on or they would be late.
the other day/night/morning/afternoon/evening
on a day/night/morning/afternoon/evening in the recent past
The other morning, I saw a deer on our lawn.
backseat driver
bed head
He is a small man with a salt-and-pepper beard, very bright brown eyes and a perpetual case of bed head, little tufts flying every which way.— Alex Witchel
city slicker
city slickers who’ve never seen a real farm
party pooper
Don’t be such a party pooper!
a party pooper who insisted they turn the music down
tear down
Tear down this wall! - Ronald Reagan
treble clef
- a clef that places G above middle C on the second line of the staff
bass clef
- a clef placing the F below middle C on the fourth line of the staff
My English Words List - November - 2025
baguette
noun
On a first-time trip to Paris, biting into the city’s best baguettes and croissants is as much on the bucket list for many as a visit to the Eiffel Tower or Louvre. — Lane Nieset, Travel + Leisure, 5 Oct. 2025
kinsman
noun
A near neighbour is better than a far-dwelling kinsman.
broom
noun
A new broom sweeps clean.
groom
noun
copycat
noun
She called me a copycat for wearing the same dress.
provoke
verb
The animal will not attack unless it is provoked.
Prime Minister Mark Carney says he advised Premier Doug Ford against running the anti-tariff ad campaign that provoked U.S. President Donald Trump and spurred him to break off trade negotiations. — Waterloo Region Record, November 3, 2025
bumpkin
noun
We enjoyed living a kind of country bumpkin way of life.
killjoy
noun
his perpetually negative attitude made him a real killjoy when others were trying to have fun
homebody
noun
He’s a homebody who hates parties.
nitpick
verb
My mom would nitpick everything about me, from my hair to my etiquette. — Bon Appétit, 14 May 2021
demolish
verb
The old factory was demolished to make way for a new parking lot.
foliage
noun
trees with colorful autumn foliage
Fall foliage experience is a feast for the eyes.
psychosis
noun
Cannabis users now develop psychosis at five times the rate of non-users. — Joseph Epstein, Newsweek, 10 Dec. 2024
veteran
noun
Veterans’ Week and Remembrance Day
Canada’s oldest veteran turns 110 on Remembrance Day week.
staff
noun
- the horizontal lines with their spaces on which music is written
itch
verb
This sweater makes me itch.
noun
I had a slight itch on my back.