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On mathematical notation Zoe O’Connor January 11, 2011 Two multiplied by an unknown number plus three equals seven. Rather tricky to read, isn’t it? This is why we use notation in maths. It’s much easier to understand 2x + 3 = 7. Much quicker to write too, and it takes up less space. I’ll assume you’re familiar with symbols such as +, ×, ÷, −, =. Some of the other notation we use can look confusing at ﬁrst, but there’s a good reason we use it! “x” This innocent letter is enough to make some people run screaming out of the lecture theatre. We use letters to denote unknown values - for instance in the equation 2x + 3 = 7, x means “an unknown quantity”. We also use letters to make a general statement: if we want to say “half of something” we write x . And the 2 choice of letter isn’t important - I could have put a or Q or even γ . (Mathematicians start using the Greek 2 2 2 alphabet when we run out of letters from the Latin alphabet. We’ve even been known to dabble with the Hebrew alphabet.) Notation to save time, eﬀort and space Here are some symbols which we use to save... well... eﬀort, mainly. Feel free to use them liberally, and mathematicians will understand what you mean. = means “does not equal” ≥ means “is greater than or equal to”, so 8 ≥ 7 and 7 ≥ 7. Similarly: ≤ means “less than or equal to”, so 2 ≤ 100, but it is not true to say that 100 ≤ 2. > means “strictly greater than”, so we don’t allow equality; you can’t say 3 > 3 but you can say 3 > 0. < means “strictly less than”, so 3000 < 4012 but it is not true to say that 3000 < 3000. ⇒ means “implies that” ∴ means “therefore” means “because” ∃ means “there exists” (We spend a lot of time proving that things exist...) ∃! means “there exists a unique” (We also spend a lot of time proving that things are unique) ∀ means “for all” . . . means “and so on” If we write anything up neatly then we try to limit the use of such symbols, but if someone’s just had a great idea and they want to write it all down before they forget it, then using symbols and abbreviations helps. A common abbreviation is “s.t.” meaning “such that”. Sets A set is a collection of things, which we call elements. We use the notation {. . .} to show that something is a set, for example: {dog, sheep, pig, tyrannosaurus rex} 1 is a set of animals; each animal is an element of our set. We have some more symbols which are to do with sets: ∈ means “is an element of” ⊆ means “is contained in” (used to say that a set is contained inside a larger set) ⊇ means “contains” I bet you’re not surprised to learn that we make sets of numbers! Here are some special number sets: N is the set of natural numbers, what you would think of as counting numbers: {1, 2, 3, . . .} Z is the set of integers, aka “whole numbers”: {. . . , −3, −2, −1, 0, 1, 2, 3, . . .} Q is the set of rational numbers - anything which can be written as a fraction (That includes integers!) R is the set of real numbers - this is pretty much anything a non-mathematician would consider to be a √ number - all rational numbers plus anything which can’t be written as a fraction, eg: 2, π = 3.14159 . . . C is the set of (unfairly named) complex numbers - anything of the form a + ib, where a and b are real numbers, and i is the square root of minus one. Think about it. Yeah, weird. Using our symbols for sets (well, one of them) we see that there’s an order of inclusion: N⊆Z⊆Q⊆R⊆C since, for example, 42 is a natural number, and it’s an integer, and a rational number (42=42/1) and a real number and a complex number. Another example: 2.6 is a rational number (and so a real number and a complex number...) since 2.6 = 13 but it isn’t an integer since there’s something after the decimal point. 5 It’s been said that mathematicians have no sense of humour. This is a common misconception, brought about by the fact that we don’t use enough exclamation marks in our writing. But there’s a good reason for this! We use the exclamation mark to mean “factorial”, which is when we multiply a natural number by every natural number smaller than it. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120 So when a mathematician sees an exclamation mark after a number, she reads it as “factorial”, and we get the following situation: How a normal person reads a facebook status: “Dennis has just turned 12!” How a mathematician reads the same thing: “Dennis has just turned 479 001 600” Dennis is old. 2