1. Arithmetic

数学英语



周维祺

Arithmetic

Properties and manipulation of numbers

Number Systems

  • Natural Numbers:

  • Integers:

  • Rational Numbers:

  • Real Numbers:

  • Complex Numbers:

  • Irrational Numbers:

Equality

  • : is equal to
    Alternatively: equals

  • : is not equal to
    Alternatively: does not equal

Inequality

  • : is greater/larger than

  • : is smaller/less than

  • : is greater/larger than or equal to
    Alternatively: is no less/smaller than

  • : is smaller/less than or equal to
    Alternatively: is no larger/greater than

Binary Operations

  • Addition\Summation：

  • Subtraction：

  • Multiplication\Product：

  • Division:

Addition/Summation

  • ： plus is (equal to)
    ： plus equals

  • : summands

  • Alternatively: add to or add to

  • Summation:

Subtraction

  • ： minus is (equal to)
    ： minus equals

  • Alternatively: subtract from

Multiplication/Product

  • ： times is (equal to)
    ： times equals

  • : multiplicands

  • If , then is a multiple of

  • Products:

Division

  • ：
    quotient:
    remainder:
    dividend:
    divisor:

Fractions

  • over or -th (less used)

  • : numerator

  • : denominator

Decimals

  • Three point one four

  • : the integer part

  • : the decimal point

  • : the decimal/fraction part

The 24 Game

  • The 24 game is an arithmetical puzzle in which the objective is to find a way to combine four integers using only basic arithmetic operations ( to get a result of 24

  • Each integer must be used exactly once

  • It is typically played with a standard 52-card deck, with integers ranging from 1 to 13

Demonstration

Round 1

Round 2

Round 3

Positive/Negative Numbers

  • : is positive

  • : is negative

  • : is non-negative

  • : is non-positive

  • : plus five

  • : minus three

Odd/Even Numbers

  • : is even

  • : is odd

Prime/Composite Numbers

  • is prime: and is only divisible by and itself

  • is composite: , , and is not prime

  • prime factorization:

Theorem (Euclid). There are infinitely many prime numbers.

Proof: Assume the contrary that there are only finitely many primes. Let be the set of all such numbers and consider now

Clearly is prime as

holds for all , but since , which is a contradiction.

Conventions

  • A theorem is a statement of importance and significance. A typical paper shall contain at least one theorem deduced by the author/authors.

  • Proofs should be mathematically and grammatically correct, logical, precise, concise and well written. In particular, pay attention to symbols, notations, and punctuations (in centered equations).

  • QED symbol

Tasks

state and prove your favourite theorem

Summary

  • Numbers and number systems

  • Comparisons and binary operations

  • Theorems and proofs

# Modulation/Congruence If $a=bq+r$ with $a,b,q\in\mathbb N$ and $r\in\{0,\ldots,b-1\}$, then $$a=r \pmod b,$$ i.e., $a$ modulo $b$ is (congruent to ) $r$ <br><br/> <!-- _class: lead