Arithmetics Arithmetic Syllabus

Arithmetic syllabus — full chapter and topic list

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

1. 1

   Number Systems

   6 topics

   • Natural Numbers
   • Whole Numbers
   • Integers
   • Rational Numbers
   • Irrational Numbers
   • Real Numbers
2. 2

   Basic Operations

   4 topics

   • Addition
   • Subtraction
   • Multiplication
   • Division
3. 3

   Properties of Numbers

   5 topics

   • Commutative Property
   • Associative Property
   • Distributive Property
   • Identity Property
   • Inverse Property
4. 4

   Factors and Multiples

   5 topics

   • Prime Numbers
   • Composite Numbers
   • Prime Factorization
   • Greatest Common Divisor (GCD)
   • Least Common Multiple (LCM)
5. 5

   Fractions

   7 topics

   • Proper Fractions
   • Improper Fractions
   • Mixed Numbers
   • Equivalent Fractions
   • Simplifying Fractions
   • Adding and Subtracting Fractions
   • Multiplying and Dividing Fractions
6. 6

   Decimals

   5 topics

   • Place Value
   • Converting Fractions to Decimals
   • Adding and Subtracting Decimals
   • Multiplying and Dividing Decimals
   • Rounding Decimals
7. 7

   Percentages

   5 topics

   • Understanding Percentages
   • Converting Fractions and Decimals to Percentages
   • Calculating Percentages
   • Percentage Increase and Decrease
   • Applications of Percentages
8. 8

   Ratios and Proportions

   5 topics

   • Understanding Ratios
   • Simplifying Ratios
   • Proportions
   • Solving Proportion Problems
   • Applications of Ratios and Proportions
9. 9

   Exponents and Radicals

   7 topics

   • Understanding Exponents
   • Laws of Exponents
   • Negative Exponents
   • Scientific Notation
   • Square Roots
   • Cube Roots
   • Higher Order Roots
10. 10

    Sequences and Series

    5 topics

    • Arithmetic Sequences
    • Geometric Sequences
    • Finding the n-th Term
    • Summation of Series
    • Applications of Sequences and Series

Arithmetic flashcards for Arithmetics

25 of 50 cards from the Arithmetic deck — real questions with worked answers.

1. 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: $\{0, 1, 2, 3, \dots\}$.)

2. What is the set of whole numbers?

   The whole numbers are the natural numbers together with zero: $\{0, 1, 2, 3, \dots\}$.

3. 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.

4. 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 $\mathbb{Q}$.

5. 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. $0.\overline{3} = \frac{1}{3}$).

6. 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.

7. Give three classic examples of irrational numbers.

   $\sqrt{2}$, $\pi$, and $e$ are all irrational.

8. 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.

9. State the subset chain relating the main number systems.

   $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, and the irrationals are $\mathbb{R} \setminus \mathbb{Q}$.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. Give a counterexample showing subtraction is not commutative.

    $5 - 3 = 2$ but $3 - 5 = -2$, so $5 - 3 \neq 3 - 5$.

17. 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.

18. State the distributive property of multiplication over addition.

    $a \times (b + c) = a \times b + a \times c$. Multiplication distributes across a sum.

19. Use the distributive property to expand $6 \times (10 + 3)$.

    $6 \times (10 + 3) = 6 \times 10 + 6 \times 3 = 60 + 18 = 78$.

20. 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.

21. What is the multiplicative identity?

    The multiplicative identity is $1$: for any number $a$, $a \times 1 = a$.

22. State the additive inverse property.

    Every number $a$ has an additive inverse $-a$ such that $a + (-a) = 0$.

23. 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.

24. What is the multiplicative inverse of $\frac{3}{5}$?

    Its reciprocal $\frac{5}{3}$, since $\frac{3}{5} \times \frac{5}{3} = 1$.

25. Define a prime number.

    A prime number is a natural number greater than $1$ that has exactly two distinct positive divisors: $1$ and itself.

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