Nutrition
A daily calorie count is one long sum: 320 + 450 + 610 + 180 = 1,560 kcal — only the total says whether you stayed inside the plan.
Addition
Household budget
Take a 4,200 payday, subtract the fixed bills — 1,350 rent, 280 electricity, 90 internet — and the 2,480 left over is what the month actually has to work with.
Subtraction
Cooking
The recipe serves two and six people are coming, so every quantity triples: 250 g of pasta becomes 750 g, and 1.5 tbsp of oil becomes 4.5 tbsp.
Multiplication
Splitting a bill
A 246 bill across four people is 61.50 each — and the fact that 246 : 4 does not come out whole is precisely where the decimal part starts to matter.
Division
A shopping basket
Three yoghurts at 4.50 and two cheeses at 7.20 come to 3 · 4.50 + 2 · 7.20 = 27.90. Work strictly left to right instead and you get 111.60 — the order is the difference between a bill and nonsense.
Order of operations
Weather forecast
At night the thermometer reads −6 °C, at noon 4 °C. The difference is 10 degrees, because it is 6 marks from −6 up to 0 and another 4 from 0 up to 4 — one subtraction, 4 − (−6), not two separate sums.
Negative numbers Convert it here: Temperature
Shopping
At a self-checkout you keep a running total in your head: 12.80 + 7.20 + 23.40 is roughly 13 + 7 + 23 = 43. When the terminal says 143, you know at once that something scanned twice.
Rounding and estimation
Renovation and buying materials
Tiles at 34 per square metre for a 26 m² floor come to 34 · 26 = 884 — the sum you do on a scrap of paper at the builders merchant counter to check that the figure on their screen is right.
Written arithmetic
Clocks and old buildings
A town-hall clock face writes four as IIII or IV and twelve as XII — and the date MDCCLXXXVIII carved above the door reads 1788, so the building is over 230 years old.
Roman numerals
A drive to see family
It is 280 km and you average 70 km/h, so 280 : 70 = 4 hours on the road. Leave at 13:30 and you arrive at 17:30 — which is exactly what to say on the phone.
Speed, distance and time Convert it here: Speed

Branch formulas

Branch: Arithmetic

Addition

Sum
$a + b = c$
addend + addend = sum
Commutativity
$a + b = b + a$
the order of addends does not change the result
Associativity
$(a + b) + c = a + (b + c)$
the grouping of addends does not change the result
Identity element
$a + 0 = a$
zero does not change the number

Subtraction

Difference
$a - b = c$
minuend − subtrahend = difference
Check
$a - b = c \iff b + c = a$
check subtraction with addition
Subtracting zero
$a - 0 = a$
zero leaves the number unchanged
Difference of equals
$a - a = 0$
a number minus itself is zero

Multiplication

Product
$a \cdot b = c$
factor · factor = product
Commutativity
$a \cdot b = b \cdot a$
the order of factors does not change the result
Associativity
$(a \cdot b) \cdot c = a \cdot (b \cdot c)$
grouping of factors does not change the result
Distributivity
$a \cdot (b + c) = a \cdot b + a \cdot c$
multiplication distributes over addition
Identity element
$a \cdot 1 = a$
multiplying by one leaves the number unchanged
Multiplying by zero
$a \cdot 0 = 0$
a product with zero is always zero

Division

Quotient
$a : b = c$
dividend : divisor = quotient (b ≠ 0)
Check
$a : b = c \iff c \cdot b = a$
check division with multiplication
Division with remainder
$a = b \cdot q + r$
the remainder satisfies 0 ≤ r < b
Dividing by one
$a : 1 = a$
dividing by one leaves the number unchanged
Dividing by itself
$a : a = 1$
for a ≠ 0

Order of operations

Multiplication before addition
$2 + 3 \cdot 4 = 14$
first 3 · 4 = 12, then 2 + 12
Brackets first
$(2 + 3) \cdot 4 = 20$
brackets change the order: first 2 + 3
Power before multiplication
$2^3 \cdot 2 = 16$
first 2³ = 8, then 8 · 2
Left to right
$20 - 4 - 3 = 13$
operations of the same rank are done left to right

Negative numbers

The opposite number
$-(-a) = a$
the opposite of the opposite is the number itself
Sum of opposites
$a + (-a) = 0$
opposite numbers cancel to zero
Subtraction as addition
$a - b = a + (-b)$
to subtract a number is to add its opposite
Subtracting a negative
$a - (-b) = a + b$
two minus signs side by side make a plus
Different signs
$(-a) \cdot b = -(a \cdot b)$
a product of numbers with different signs is negative
Same signs
$(-a) \cdot (-b) = a \cdot b$
a product of two negative numbers is positive

Rounding and estimation

Rounded to tens
$63 \approx 60$
the units digit is 3, so down
Rounded to hundreds
$2847 \approx 2800$
the tens digit is 4, so down
Rounded to thousands
$2847 \approx 3000$
the hundreds digit is 8, so up
Estimating a sum
$487 + 216 \approx 490 + 220 = 710$
round the terms, then add
Estimating a product
$38 \cdot 21 \approx 40 \cdot 20 = 800$
round the factors, then multiply

Written arithmetic

Addition with a carry
$\begin{array}{r} \overset{1}{4}\overset{1}{7}8 \\ +\;356 \\ \hline 834 \end{array}$
the small ones above a column are the carries
Subtraction with a borrow
$\begin{array}{r} 8124 \\ \underline{-\;2758} \\ 5366 \end{array}$
we borrow a ten from the column on the left
Multiplication with partial products
$\begin{array}{r} 347 \\ \times\;26 \\ \hline 2082 \\ \underline{694\phantom{0}} \\ 9022 \end{array}$
the second product shifts one place to the left
Long division
$\begin{array}{r} 123 \\ 4\,)\overline{\,492} \\ \underline{4\phantom{92}} \\ 09\phantom{2} \\ \underline{\phantom{4}8\phantom{2}} \\ \phantom{4}12 \\ \underline{\phantom{4}12} \\ \phantom{49}0 \end{array}$
divide, multiply, subtract, bring down the next digit

Roman numerals

The seven signs
$\text{I} = 1 \;\; \text{V} = 5 \;\; \text{X} = 10 \;\; \text{L} = 50 \;\; \text{C} = 100 \;\; \text{D} = 500 \;\; \text{M} = 1000$
the whole alphabet of the Roman system
The additive rule
$\text{XXVII} = 10 + 10 + 5 + 1 + 1 = 27$
signs in non-increasing order are added
Subtractive spelling
$\text{XIV} = 10 + (5 - 1) = 14$
a smaller sign before a larger one is subtracted
A year on a foundation stone
$\text{MCMXCIV} = 1000 + 900 + 90 + 4 = 1994$
CM = 900, XC = 90, IV = 4

Speed, distance and time

Speed
$v = \frac{s}{t}$
distance divided by time
Distance
$s = v \cdot t$
speed times time
Time
$t = \frac{s}{v}$
distance divided by speed
Units of time
$1\,\text{h} = 60\,\text{min} = 3600\,\text{s}$
a clock counts in sixties, not in hundreds
Converting speed units
$1\,\frac{\text{m}}{\text{s}} = 3.6\,\frac{\text{km}}{\text{h}}$
from m/s to km/h, multiply by 3.6

Plus, minus, times and division signs and digits floating in a geometric grid, joined by emerald lines of the order of operations, against a cosmic nebula in violet and magenta

Arithmetic

The grammar of arithmetic — why the order of operations is a convention

Nobody invented +, −, × and ÷ overnight, and the fight over 6 ÷ 2(1+2) still splits the internet. Here is why the notation of arithmetic is a human convention, the results are absolute truth, and division by zero simply cannot be defined.

10 min