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Arithmetics Syllabus

Mathematical study and application. The full Arithmetics syllabus: 10 chapters and 54 topics, with 50 flashcards written against it.

10Chapters

54Topics

50Flashcards

Arithmetics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Arithmetics, not a summary of it.

Number Systems
6 topics
- Natural Numbers
- Whole Numbers
- Integers
- Rational Numbers
- Irrational Numbers
- Real Numbers
Basic Operations
4 topics
- Addition
- Subtraction
- Multiplication
- Division
Properties of Numbers
5 topics
- Commutative Property
- Associative Property
- Distributive Property
- Identity Property
- Inverse Property
Factors and Multiples
5 topics
- Prime Numbers
- Composite Numbers
- Prime Factorization
- Greatest Common Divisor (GCD)
- Least Common Multiple (LCM)
Fractions
7 topics
- Proper Fractions
- Improper Fractions
- Mixed Numbers
- Equivalent Fractions
- Simplifying Fractions
- Adding and Subtracting Fractions
- Multiplying and Dividing Fractions
Decimals
5 topics
- Place Value
- Converting Fractions to Decimals
- Adding and Subtracting Decimals
- Multiplying and Dividing Decimals
- Rounding Decimals
Percentages
5 topics
- Understanding Percentages
- Converting Fractions and Decimals to Percentages
- Calculating Percentages
- Percentage Increase and Decrease
- Applications of Percentages
Ratios and Proportions
5 topics
- Understanding Ratios
- Simplifying Ratios
- Proportions
- Solving Proportion Problems
- Applications of Ratios and Proportions
Exponents and Radicals
7 topics
- Understanding Exponents
- Laws of Exponents
- Negative Exponents
- Scientific Notation
- Square Roots
- Cube Roots
- Higher Order Roots
Sequences and Series
5 topics
- Arithmetic Sequences
- Geometric Sequences
- Finding the n-th Term
- Summation of Series
- Applications of Sequences and Series

Arithmetics flashcards

The complete Arithmetics deck: 50 questions with worked answers.

What is the set of natural numbers, and how is it denoted?
The natural numbers are the counting numbers $\mathbb{N} = \{1, 2, 3, 4, \dots\}$. (In some conventions $0$ is included: $\{0, 1, 2, 3, \dots\}$.)
What is the set of whole numbers?
The whole numbers are the natural numbers together with zero: $\{0, 1, 2, 3, \dots\}$.
What distinguishes integers from whole numbers?
Integers include the whole numbers plus their negatives: $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. Whole numbers have no negatives.
Define a rational number and give its symbol.
A rational number is any number expressible as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$. The set is denoted $\mathbb{Q}$.
How can you recognize a rational number from its decimal expansion?
A rational number has a decimal expansion that either terminates (e.g. $0.75$) or repeats periodically (e.g. $0.\overline{3} = \frac{1}{3}$).
Define an irrational number.
An irrational number is a real number that cannot be written as $\frac{p}{q}$ for integers $p, q$. Its decimal expansion is non-terminating and non-repeating.
Give three classic examples of irrational numbers.
$\sqrt{2}$, $\pi$, and $e$ are all irrational.
What is the set of real numbers?
The real numbers $\mathbb{R}$ are all rational and irrational numbers together — every point on the number line.
State the subset chain relating the main number systems.
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, and the irrationals are $\mathbb{R} \setminus \mathbb{Q}$.
Is $\sqrt{9}$ rational or irrational? Explain.
Rational, because $\sqrt{9} = 3$, which is an integer. Only roots of non-perfect-squares (like $\sqrt{2}$) are irrational.
What are the names of the two numbers being added in an addition, and the result?
The numbers being added are the addends (or summands); the result is the sum.
In subtraction $a - b = c$, what are $a$, $b$, and $c$ called?
$a$ is the minuend, $b$ is the subtrahend, and $c$ is the difference.
In multiplication, what are the input numbers and the result called?
The numbers multiplied are the factors (or multiplicand and multiplier); the result is the product.
In division $a \div b = c$, name $a$, $b$, and $c$.
$a$ is the dividend, $b$ is the divisor, and $c$ is the quotient. Any leftover is the remainder.
State the commutative property and which operations obey it.
Commutativity means order does not affect the result: $a + b = b + a$ and $a \times b = b \times a$. It holds for addition and multiplication, but not for subtraction or division.
Give a counterexample showing subtraction is not commutative.
$5 - 3 = 2$ but $3 - 5 = -2$, so $5 - 3 \neq 3 - 5$.
State the associative property.
Grouping does not affect the result: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$. Holds for addition and multiplication only.
State the distributive property of multiplication over addition.
$a \times (b + c) = a \times b + a \times c$. Multiplication distributes across a sum.
Use the distributive property to expand $6 \times (10 + 3)$.
$6 \times (10 + 3) = 6 \times 10 + 6 \times 3 = 60 + 18 = 78$.
What is the additive identity, and what property defines it?
The additive identity is $0$: for any number $a$, $a + 0 = a$. Adding zero leaves a number unchanged.

Show the other 30 Arithmetics flashcards

What is the multiplicative identity?
The multiplicative identity is $1$: for any number $a$, $a \times 1 = a$.
State the additive inverse property.
Every number $a$ has an additive inverse $-a$ such that $a + (-a) = 0$.
State the multiplicative inverse (reciprocal) property.
Every nonzero number $a$ has a multiplicative inverse $\frac{1}{a}$ such that $a \times \frac{1}{a} = 1$. Zero has no multiplicative inverse.
What is the multiplicative inverse of $\frac{3}{5}$?
Its reciprocal $\frac{5}{3}$, since $\frac{3}{5} \times \frac{5}{3} = 1$.
Define a prime number.
A prime number is a natural number greater than $1$ that has exactly two distinct positive divisors: $1$ and itself.
List the first eight prime numbers.
$2, 3, 5, 7, 11, 13, 17, 19$.
Why is $1$ neither prime nor composite?
$1$ has only one positive divisor (itself), so it fails the two-divisor requirement for primes and the more-than-two requirement for composites.
What is the only even prime number?
$2$ is the only even prime; every other even number is divisible by $2$ and thus composite.
Define a composite number.
A composite number is a natural number greater than $1$ that has more than two positive divisors — i.e., it has at least one factor other than $1$ and itself.
Is $51$ prime or composite? Justify.
Composite, because $51 = 3 \times 17$, so it has divisors beyond $1$ and itself.
What is prime factorization?
Prime factorization is expressing a composite number as a product of prime numbers, e.g. $60 = 2^{2} \times 3 \times 5$.
State the Fundamental Theorem of Arithmetic.
Every integer greater than $1$ can be written as a product of primes in exactly one way, apart from the order of the factors.
Find the prime factorization of $84$.
$84 = 2^{2} \times 3 \times 7$.
How do you find the GCD of two numbers using prime factorization?
Factor both into primes and multiply the common prime powers using the lowest exponent of each. E.g. $12 = 2^{2}\times 3$, $18 = 2\times 3^{2}$, so $\gcd = 2^{1}\times 3^{1} = 6$.
How do you find the LCM using prime factorization?
Take each prime that appears and use its highest exponent. E.g. $12 = 2^{2}\times 3$, $18 = 2\times 3^{2}$, so $\operatorname{lcm} = 2^{2}\times 3^{2} = 36$.
What is the result of adding two integers with opposite signs?
Subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value, e.g. $-7 + 3 = -4$.
What is the rule for the sign of a product/quotient of two numbers?
Same signs give a positive result; different signs give a negative result. E.g. $(-4)\times(-3)=12$ and $(-4)\times 3 = -12$.
How does subtracting a number relate to addition?
Subtracting a number equals adding its additive inverse: $a - b = a + (-b)$.
How does dividing by a fraction relate to multiplication?
Dividing by a fraction equals multiplying by its reciprocal: $a \div \frac{b}{c} = a \times \frac{c}{b}$.
Why is division by zero undefined?
$a \div 0$ asks for a number $x$ with $0 \times x = a$; for $a \neq 0$ no such $x$ exists, and for $a = 0$ every $x$ works, so it is undefined.
Is the set of integers closed under division? Give an example.
No. For example $7 \div 2 = 3.5$, which is not an integer. Integers are closed under addition, subtraction, and multiplication only.
State the standard order of operations.
Parentheses, then Exponents, then Multiplication and Division (left to right), then Addition and Subtraction (left to right) — PEMDAS.
Evaluate $2 + 3 \times 4^{2}$ using order of operations.
$4^{2} = 16$, then $3 \times 16 = 48$, then $2 + 48 = 50$.
What is a divisor (factor) of a number?
A divisor of $n$ is an integer that divides $n$ evenly, leaving remainder $0$. E.g. the divisors of $6$ are $1, 2, 3, 6$.
State the divisibility rule for $3$.
A number is divisible by $3$ if the sum of its digits is divisible by $3$. E.g. $123$: $1+2+3=6$, divisible by $3$.
State the divisibility rule for $9$.
A number is divisible by $9$ if the sum of its digits is divisible by $9$. E.g. $729$: $7+2+9=18$, divisible by $9$.
What does it mean for a number to be even or odd, in terms of divisibility?
An even number is divisible by $2$ (ends in $0,2,4,6,8$); an odd number is not divisible by $2$ (ends in $1,3,5,7,9$).
Compare the density of rationals and integers on the number line.
Integers are discrete (gaps of $1$ between them), while rationals are dense — between any two rationals there is always another rational.
Classify the number $0$ across the number systems.
$0$ is a whole number, an integer, a rational number ($\frac{0}{1}$), and a real number, but it is not a natural number (in the counting convention) and is neither prime nor composite.
What is the sum of an irrational number and a nonzero rational number?
It is always irrational. E.g. $2 + \sqrt{3}$ is irrational, because if it were rational, subtracting the rational $2$ would make $\sqrt{3}$ rational — a contradiction.

Arithmetics syllabus FAQ

What is in the Arithmetics syllabus?

The Arithmetics syllabus (Arithmetic) is split into 10 chapters: Number Systems, Basic Operations, Properties of Numbers, Factors and Multiples, Fractions, Decimals, Percentages, Ratios and Proportions, Exponents and Radicals and Sequences and Series. Together they contain 54 topics.

How many chapters and topics are in the Arithmetics syllabus?

10 chapters and 54 topics. The largest chapter is Fractions, with 7 topics.

Are there Arithmetics flashcards?

Yes — 50 question-and-answer flashcards, printed in full on this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.

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