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MATH 2UU3 * GLOSSARY of TERMS Blue: Covered So Far Black: to Be Covered

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MATH 2UU3 * GLOSSARY of TERMS Blue: Covered So Far Black: to Be Covered MATH 2UU3 * GLOSSARY OF TERMS blue: covered so far black: to be covered Quantitative reasoning: Applying and correct logical reasoning about math symbols, objects, calculations, algorithms, ideas in areas outside mathematics (meaning in context which is not abstract mathematics). Prime number: A positive integer that has exactly two divisors; 1 and itself. [we will not talk about prime numbers; this was to illustrate the concept of a math definition, so no need to memorize this definition] Definition: In math, and elsewhere (e.g., legal documents, financial agreements), it introduces something new (new term, concept, idea, object) based on previously defined terms, concepts, ideas or objects. Math definitions are precise, clear, unambiguous, and universal (do not depend on culture, political system, human condition, etc.). Once established they very rarely ever change. Red herring: An expression which is misleading, confusing, or created to be misleading or confusing. It could also be a sentence, or a statement, which, by introducing certain sub-plot(s), draws the attention away from the actual issue or problem. Correlation: Existence of a noticeable pattern (also called relationship, or trend, or association) between two quantities (or variables). For instance, if input/explanatory variable and output/ response variables both increase, or one increases when the other decreases (say over time), they are said to be correlated. Note: correlation does not mean causation. Just because a correlation exists does not mean that changes in one variable actually cause changes in the other variable. This might be the case, but there could also be a third variable that is causing the changes in both other variables; as well, there could be other possibilities, including coincidence. Positive correlation: One quantity (variable) increases when the other increases. Or, one quantity (variable) decreases when the other decreases. Negative correlation: One quantity (variable) decreases when the other increases. Theorem: A mathematical statement that has been rigorously shown to be true (i.e., “proven true”) through logical reasoning from a set of axioms, or from previously established (proven) theorems. In other words proof is mathematically acceptable evidence. See implication. Implication: a logical statement of the form “IF (assumption/s) THEN (conclusion/s).” This is the form of every theorem. An implication is valid (i.e., can be used) only if all assumption/s is/are true. If even one assumption fails to hold, the implication (theorem) cannot be used. However, the conclusions/s might still be true, but they do not follow (are not caused) by the implication. Causation: outside mathematics, an implication is known as causation i.e., as a sentence that establishes a causal relationship. Instead of “A implies B,” we say “A causes B,” “B is a consequence of A,” “B is caused by A,” “B happened because of A,” “A is a reason for B,” etc. Converse: The converse of the implication “IF A THEN B” is “IF B THEN A.” The converse of a true implication is not necessarily true. Contrapositive: The contrapositive of the implication “IF A THEN B” is “IF NOT B THEN NOT A.” The contrapositive of a true implication is always true, and the contrapositive of a false implication is always false. Equivalence: If A causes B and B causes A then we say that A is equivalent to B. In other words, IF A THEN B and its converse IF B THEN A are both true, then A is equivalent to B. Negation: a logical structure; is we say that B is NOT A means that B is anything that is not A; for instance if B=NOT a cat, then B can be a dog, a chair, a planet, DNA molecule, etc. Trend, association: see correlation. Quantifier: Specifies elements of a given set which have some property. Universal quantifier (”for all”) signifies that all elements in a given set have that property. Existential quantifier (“there exists,” or “there is”) signifies that at least one element in a given set has that property. Hypothesis: see conjecture Conjecture or hypothesis: statement that is formulated based on identifying a pattern, or supported by data; however, there is no definitive proof of its validity. Once such proof is established, a conjecture becomes a theorem. Replicability: Obtaining same, or very similar results as outcomes of a repeated experiment. In other words, an experiment is established as having a valid conclusion if its results can be replicated. Randomized control trial: is a study/ experiment, in which participants are divided randomly into two groups: treatment group (that is under some kind of intervention, e.g., given a pill that is supposed to reduce pain) and control group (that is not under intervention, and is often given a placebo, e.g., a pill that does nothing). Based on the differences in the reactions of the two groups, and through statistical analysis, one can establish whether or not the intervention is effective. Meta-analysis: a summary, or a critical analysis, of multiple studies. Axiom: A mathematical fact (or a fact in general) that we take for granted, to start a math theory (or other scientific theory); thus, we cannot prove that an axiom is true (e.g. of an axiom from calculus: x*y=y*x for any two real numbers x and y) Algorithm: A step by step process or set of rules for accomplishing some task. An example was given in class: Luhn algorithm that verifies whether or not a given number is a legitimate credit card number. Estimate: A calculation or an algorithm that involves making assumptions about something that cannot be directly measured (or is hard or impractical to measure), or when exact value of some data is not (or could not have been) known. When making an estimate, we need to write down all assumptions that we made, and list the sources of these assumptions. Approximation: A value that is nearly exact, or close to a true value (which might, or might not be known, or knowable). Scientific notation: A method of writing concisely by using powers of ten (e.g. 0.00278 = 2.87*10^-3, 10000 = 1.0*10^4). Convenient for very small and very large numbers. To write a number in scientific notation means to write it in the form D.dddd * 10 ^ exponent, where D is a non-zero digit, dddd are decimals and the exponent could be positive, negative, or zero. Order of magnitude: Very rough way to compare numbers. It refers to the powers of ten in scientific notation. For example, 4.7*10^7 has order of magnitude 7, and is one order of magnitude bigger than, say, 3.9*10^6. The number 4.3*10^-5 has order of magnitude -5. Nanoscale technology: manufacturing objects (or structures) whose size is about a nanometre, i.e., one billionth of a metre. Absolute number: A number quoted as is, without reference to anything else (for example: I paid $2.45 for bananas; there are about 500 thousand people in Hamilton) Relative number: A number related to something else (for instance price per unit: bananas cost $0.69 per pound; or comparison: by population size, Hamilton is 10th largest city in Canada) Absolute change: A direct comparison of the values of two quantities, computed by subtraction (by how much is one quantity larger than the other quantity?) In particular, if some quantity changes from A to B, then the absolute change is B-A. Relative change: A comparison of two quantities by expressing one as proportion (or percent) of the other. In particular, if some quantity changes from A to B, then the relative change (relative with respect to the initial value) is (B-A)/A. Positional number system: number system where the position of a digit determines its value; for instance, 4 in 3456 represents 4 hundred, whereas 4 in 2249 is forty (four tens). Example of a non-positional system are roman numerals, where the same symbol always has the same value. For Instance, all Cs in MCCCXIX and all C’s in DCCV have the same value of 100. Decimal number system: Number system that we use daily, based on powers of 10, which uses Hindu-Arabic numerals 0, 1, 2, 3, …, 8, 9. Binary number system: Number system based on powers of 2; uses digits 0 and 1; use in computers, data transmission, etc. Hexadecimal number system: : Number system based on powers of 16; uses digits 0, 1, 2, ..., 9, and letter equivalents A, B, C, D, E, F for 10, 11, 12, 13, 14, 15 respectively; use in computers, data transmission, etc. Vigesimal number system: Number system based on powers of 20; uses pictorial representations for digits 0, 1, 2, 3, …, 19. Still, in some forms, in sporadic use. Sexagesimal number system: Number system based on powers of 60; uses pictorial representations for digits 0, 1, 2, 3, …, 59. This is why an hour has 60 minutes and a minute has 60 seconds. (and also why the right angle has 90 degrees). Buy rate: In currency exchange, a buy rate (common meaning) is the rate at which a bank buys currency from individuals. Sometimes, however, it’s the rate at which an individual buys a currency from a bank. To figure which of the rates posted is a buy rate, we keep in mind is that it is the one that works against us. Sell rate: In currency exchange, a sell rate (common meaning) is the rate at which a bank sells currency to individuals. Sometimes, however, it’s the rate at which an individual sells a currency to a bank. To figure which of the rates is a buy rate, we keep in mind is that it is the one that works against us.
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