1. Numbers and place value
Arithmetic studies numbers and the basic operations used to combine, compare, and transform them.
The main arithmetic operations are:
Addition
Subtraction
Multiplication
Division
Number sets
| Number set | Description | Examples |
|---|---|---|
| Natural numbers | Counting numbers | $1,\ 2,\ 3,\ 4$ |
| Whole numbers | Natural numbers and zero | $0,\ 1,\ 2,\ 3$ |
| Integers | Positive numbers, negative numbers, and zero | $-3,\ -2,\ -1,\ 0,\ 1$ |
| Rational numbers | Numbers expressible as a fraction | $\frac{1}{2},\ -\frac{5}{3},\ 0.75$ |
| Irrational numbers | Numbers not expressible as a terminating or repeating fraction | $\sqrt{2},\ \pi$ |
| Real numbers | Rational and irrational numbers together | $-4,\ 0,\ \frac{2}{3},\ \sqrt{5}$ |
Arithmetic usually begins with whole numbers, then extends to integers, fractions, decimals, and real numbers.
Place value
In base ten, each digit's value depends on its position.
For the number:
the place values are:
| Digit | Place | Value |
|---|---|---|
| $4$ | Thousands | $4000$ |
| $3$ | Hundreds | $300$ |
| $8$ | Tens | $80$ |
| $2$ | Ones | $2$ |
| $6$ | Tenths | $0.6$ |
| $5$ | Hundredths | $0.05$ |
Expanded form:
Comparing numbers
Use the symbols:
| Symbol | Meaning |
|---|---|
| $=$ | equal to |
| $\ne$ | not equal to |
| $<$ | less than |
| $>$ | greater than |
| $\le$ | less than or equal to |
| $\ge$ | greater than or equal to |
Examples:
Absolute value
The absolute value of a number is its distance from zero.
Examples:
2. Addition
Addition combines quantities.
For two numbers:
where:
$a$ and $b$ are addends
$c$ is the sum
Properties of addition
| Property | Rule |
|---|---|
| Commutative property | $a + b = b + a$ |
| Associative property | $(a + b) + c = a + (b + c)$ |
| Additive identity | $a + 0 = a$ |
| Additive inverse | $a + (-a) = 0$ |
Adding whole numbers
Line up digits by place value.
Example:
Add ones:
Write $4$ and carry $1$.
Add tens:
Write $2$ and carry $1$.
Add hundreds:
Therefore:
Adding decimals
Line up decimal points before adding.
Example:
Rewrite as:
Then:
Adding like terms
Only quantities with the same units or same type should be added directly.
Examples:
Unlike quantities cannot be combined without conversion or interpretation.
3. Subtraction
Subtraction finds the difference between quantities.
For two numbers:
where:
$a$ is the minuend
$b$ is the subtrahend
$c$ is the difference
Subtraction can be rewritten as addition of the opposite:
Properties of subtraction
Subtraction is not commutative:
Example:
but
Subtraction is not associative:
Subtracting whole numbers
Line up digits by place value.
Example:
Since $2 < 8$, borrow from the tens place. Since the tens digit is $0$, borrow from the hundreds place first.
Then:
Therefore:
Subtracting decimals
Line up decimal points.
Example:
Rewrite as:
Then:
Difference as distance
The distance between two numbers $a$ and $b$ on a number line is:
Example:
4. Multiplication
Multiplication represents repeated addition, scaling, or area.
For two numbers:
where:
$a$ and $b$ are factors
$c$ is the product
Equivalent notation:
Properties of multiplication
| Property | Rule |
|---|---|
| Commutative property | $ab = ba$ |
| Associative property | $(ab)c = a(bc)$ |
| Multiplicative identity | $a \cdot 1 = a$ |
| Multiplicative property of zero | $a \cdot 0 = 0$ |
| Distributive property | $a(b+c)=ab+ac$ |
Repeated addition
For a whole number $n$:
Example:
Multiplying by powers of ten
Multiplying by $10^n$ shifts the decimal point $n$ places to the right.
Examples:
Area interpretation
The area of a rectangle is:
where:
$l$ = length
$w$ = width
Example:
5. Division
Division separates a quantity into equal groups or finds a rate.
For two numbers:
where:
$a$ is the dividend
$b$ is the divisor
$c$ is the quotient
Division can also be written as:
Division by zero is undefined:
Division and multiplication
Division is the inverse of multiplication.
If
then
Example:
because
Remainders
For whole-number division:
where:
$q$ = quotient
$r$ = remainder
$0 \le r < b$
Example:
because:
Dividing by powers of ten
Dividing by $10^n$ shifts the decimal point $n$ places to the left.
Examples:
Fractions as division
A fraction represents division:
where
Example:
6. Order of operations
Order of operations gives a consistent way to evaluate expressions.
Use this sequence:
Parentheses and grouping symbols
Exponents and roots
Multiplication and division from left to right
Addition and subtraction from left to right
A common memory aid is PEMDAS, but multiplication and division have equal priority. Addition and subtraction also have equal priority.
Grouping symbols
Evaluate inside grouping symbols first.
Common grouping symbols:
Example:
Without parentheses:
Left-to-right rule
For operations with equal priority, work left to right.
Example:
Evaluate left to right:
Therefore:
Nested expressions
Example:
First:
and
Then:
7. Integers and signed arithmetic
Integers include positive numbers, negative numbers, and zero.
Number line
Numbers to the right are greater.
Numbers to the left are smaller.
Example:
because $-2$ is to the right of $-5$.
Interactive visual
Number line moves
Change a starting value and a step to see addition and subtraction as movement on a number line.
Adding signed numbers
Same signs: add absolute values and keep the sign.
Examples:
Different signs: subtract absolute values and keep the sign of the number with larger absolute value.
Examples:
Subtracting signed numbers
Rewrite subtraction as addition of the opposite:
Examples:
Multiplying and dividing signed numbers
| Signs | Product or quotient |
|---|---|
| Positive and positive | Positive |
| Negative and negative | Positive |
| Positive and negative | Negative |
| Negative and positive | Negative |
Examples:
Opposites
The opposite of $a$ is $-a$.
The sum of opposites is zero:
8. Factors, multiples, and primes
Factors
A factor divides a number evenly.
If
then $b$ and $c$ are factors of $a$.
Example:
so $6$ and $4$ are factors of $24$.
Multiples
A multiple of a number is the product of that number and an integer.
Multiples of $6$ include:
Prime and composite numbers
A prime number has exactly two positive factors: $1$ and itself.
Examples:
A composite number has more than two positive factors.
Examples:
The number $1$ is neither prime nor composite.
Prime factorization
Prime factorization writes a number as a product of primes.
Example:
Greatest common factor
The greatest common factor, or GCF, is the largest factor shared by two or more numbers.
Example:
Factors of $18$:
Factors of $24$:
Therefore:
Least common multiple
The least common multiple, or LCM, is the smallest positive multiple shared by two or more numbers.
Example:
Multiples of $6$:
Multiples of $8$:
Therefore:
9. Fractions
A fraction represents part of a whole, a ratio, or division.
where:
$a$ is the numerator
$b$ is the denominator
$b \ne 0$
Equivalent fractions
Multiplying numerator and denominator by the same nonzero number gives an equivalent fraction.
where
Example:
Simplifying fractions
Divide numerator and denominator by their greatest common factor.
Example:
Since
then:
Multiplying fractions
Multiply numerators and multiply denominators.
where
Example:
Dividing fractions
To divide by a fraction, multiply by its reciprocal.
where
Example:
Adding and subtracting fractions
Fractions need a common denominator before addition or subtraction.
For unlike denominators:
Example:
Mixed numbers
A mixed number contains a whole number and a fraction.
Example:
Convert to an improper fraction:
Convert an improper fraction to a mixed number:
because
10. Decimals
Decimals are base-ten fractions.
Examples:
Decimal place values
| Place | Fraction | Decimal example |
|---|---|---|
| Tenths | $\frac{1}{10}$ | $0.1$ |
| Hundredths | $\frac{1}{100}$ | $0.01$ |
| Thousandths | $\frac{1}{1000}$ | $0.001$ |
| Ten-thousandths | $\frac{1}{10000}$ | $0.0001$ |
Adding and subtracting decimals
Line up decimal points.
Example:
Example:
Multiplying decimals
Multiply as whole numbers, then place the decimal according to the total number of decimal places in the factors.
Example:
Ignore decimals first:
There are three total decimal places, so:
Dividing decimals
Move the decimal in the divisor to make it a whole number. Move the decimal in the dividend by the same number of places.
Example:
Multiply both by $100$:
Therefore:
Converting decimals to fractions
Write the decimal over the correct power of ten, then simplify.
Example:
Converting fractions to decimals
Divide numerator by denominator.
Example:
11. Ratios, rates, and proportions
Ratio
A ratio compares two quantities by division.
A ratio can be written as:
or
where
Example:
Rate
A rate compares quantities with different units.
Example:
Unit rate:
Proportion
A proportion states that two ratios are equal.
where
Cross products are equal:
Solving a proportion
Example:
Cross multiply:
Scale factor
A scale factor multiplies every corresponding length.
If a figure is scaled by factor $k$, then:
If $k>1$, the figure is enlarged.
If $0<k<1$, the figure is reduced.
12. Percents
Percent means per hundred.
Examples:
Percent of a number
To find $p\%$ of a number $N$:
Example:
Finding the percent
To find what percent $A$ is of $B$:
Example:
Percent increase
Percent increase is:
Example:
A value changes from $40$ to $50$.
Percent decrease
Percent decrease is:
Example:
A value changes from $80$ to $60$.
Simple interest
Simple interest is:
where:
$I$ = interest
$P$ = principal
$r$ = annual interest rate as a decimal
$t$ = time in years
Total amount:
or
13. Exponents, roots, and scientific notation
Exponents
An exponent indicates repeated multiplication.
where $n$ is a positive integer.
Example:
Exponent rules
| Rule | Formula |
|---|---|
| Product of powers | $a^m a^n = a^{m+n}$ |
| Quotient of powers | $\frac{a^m}{a^n}=a^{m-n}$ |
| Power of a power | $(a^m)^n=a^{mn}$ |
| Power of a product | $(ab)^n=a^n b^n$ |
| Power of a quotient | $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$ |
| Zero exponent | $a^0=1$ |
| Negative exponent | $a^{-n}=\frac{1}{a^n}$ |
Restrictions:
when a variable appears in a denominator or a zero exponent.
Roots
A square root reverses squaring.
means
Example:
A cube root reverses cubing.
means
Example:
Squares and square roots
Common perfect squares:
| $n$ | $n^2$ |
|---|---|
| $1$ | $1$ |
| $2$ | $4$ |
| $3$ | $9$ |
| $4$ | $16$ |
| $5$ | $25$ |
| $6$ | $36$ |
| $7$ | $49$ |
| $8$ | $64$ |
| $9$ | $81$ |
| $10$ | $100$ |
| $11$ | $121$ |
| $12$ | $144$ |
Scientific notation
Scientific notation writes numbers in the form:
where:
and $n$ is an integer.
Examples:
Operations with scientific notation
Multiplication:
Division:
Example:
14. Estimation, rounding, and error checks
Estimation gives a quick approximate answer before or after exact calculation.
Rounding
To round to a place value:
Locate the target place.
Check the digit to its right.
If the digit is $5$ or greater, round up.
If the digit is less than $5$, keep the target digit unchanged.
Example:
Round $48.376$ to the nearest hundredth.
The hundredths digit is $7$.
The next digit is $6$, so round up:
Compatible numbers
Compatible numbers are numbers chosen to make mental arithmetic easier.
Example:
Estimating products
Round factors before multiplying.
Example:
Exact value:
The estimate is reasonable.
Estimating quotients
Choose nearby values that divide easily.
Example:
Error checks
Use inverse operations:
| Operation | Inverse check |
|---|---|
| Addition | Subtraction |
| Subtraction | Addition |
| Multiplication | Division |
| Division | Multiplication |
Examples:
If
then check:
If
then check:
15. Problem-solving workflow
Use this checklist for most arithmetic problems.
Step 1: Identify the goal
Decide what the problem asks for.
Examples:
Sum
Difference
Product
Quotient
Percent
Ratio
Missing value
Estimate
Step 2: List knowns and unknowns
Write the given values.
Assign a symbol for the unknown quantity if needed.
Example:
Step 3: Choose the operation
Use the wording and structure of the problem.
| Wording | Common operation |
|---|---|
| total, altogether, combined | Addition |
| difference, left, fewer, how much more | Subtraction |
| groups of, times, product, area | Multiplication |
| per, each, quotient, shared equally | Division |
| out of 100, discount, markup, tax | Percent |
| same ratio, scale, equivalent rates | Proportion |
Step 4: Set up the expression
Write the expression before calculating.
Example:
A $15\%$ tip on a $\$42$ bill is:
Step 5: Calculate carefully
Follow order of operations.
Line up place values when adding or subtracting.
Use common denominators for fraction addition or subtraction.
Step 6: Check units and meaning
Ask:
Does the answer need units?
Should the answer be positive?
Is the result larger or smaller than the starting value?
Is the estimate close to the exact answer?
Step 7: Verify with an inverse operation
Use the inverse operation when possible.
Examples:
16. Formula sheet
Place value and comparison
Expanded form:
Absolute value:
Distance between two numbers:
Basic operation properties
Addition:
Multiplication:
Distributive property:
Additive identity:
Multiplicative identity:
Additive inverse:
Subtraction and division
Subtraction as addition:
Division as a fraction:
where
Remainder form:
where
Signed numbers
Same signs in multiplication or division give a positive result:
Different signs in multiplication or division give a negative result:
Fractions
Equivalent fractions:
Multiplication:
Division:
Addition with common denominator:
Subtraction with common denominator:
Unlike denominators:
Decimals
Decimal to fraction:
Multiplying by powers of ten:
Dividing by powers of ten:
Ratios and proportions
Ratio:
Proportion:
Cross products:
Scale factor:
Percents
Percent as decimal:
Percent of a number:
Finding percent:
Percent increase:
Percent decrease:
Simple interest:
Total amount:
Exponents and roots
Repeated multiplication:
Product of powers:
Quotient of powers:
Power of a power:
Zero exponent:
Negative exponent:
Square root:
Scientific notation:
where
Common mistakes to avoid
Adding or subtracting decimals without lining up decimal points.
Treating subtraction as commutative.
Treating division as commutative.
Dividing by zero.
Forgetting to work left to right for multiplication/division or addition/subtraction.
Adding fractions before finding a common denominator.
Multiplying fractions by adding numerators and denominators.
Dividing fractions without using the reciprocal of the divisor.
Dropping negative signs during multi-step calculations.
Confusing percent form with decimal form.
Using $25$ instead of $0.25$ for $25\%$ in calculations.
Moving the decimal in only one number when dividing decimals.
Rounding too early in a multi-step problem.
Forgetting that $1$ is neither prime nor composite.
Assuming an estimate must equal the exact answer.
Sources
Stewart, Calculus: Early Transcendentals
Lay, Linear Algebra and Its Applications
Rosen, Discrete Mathematics and Its Applications
Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
Blitzstein and Hwang, Introduction to Probability