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Working with Quantifiers Let’s see if we can find an answer to these problems. 1) Is it true that for every real number x, x+1 is greater than or equal to x? Yes, it is. 2) Is it true there is a real number x? Well, if we were talking about the rational numbers, the…

IGCSE Mathematics: Cosine Rule Cosine Rule The cosine is just like the sine rule, but it’s for when you know 3 sides and want to find an angle OR when you know 2 sides + 1 angle (none are opposite each other meaning no sine rule) and need to find the remaining side. The cosine…

Fracons, Decimals, Percent and Raos Fractions / Rational Numbers mn, where m and n are integers and n ≠0 , is called a rational number Reducing To reduce a rational number to the lowest terms, divide the numerator and the denominator by the greatest common factor (GCF).Example 1: a) 1624=16÷ 824 ÷ 8=23b) -9-36=936=9 ÷…

Introduction to Mathematical Thinking. Proofs What is a proof? Fermat’s Last Theorem If n≥3 the equation x^n + y^n = z^n has no solutions for positive whole numbers x, y, z. Goldbach Conjecture Every even number greater than 2 can be expressed as a sum of teo primes. In mathematics, a proof is a logically…

CFA Level 2 – Quantitative Analysis, Session 3 – Reading 13, Time-Series Analysis-LOS g (Practice Questions, Sample Questions)  1. Which of the following statements regarding an out-of-sample forecast is least accurate? A) There is more error associated with out-of-sample forecasts, as compared to in-sample forecasts.B) Forecasting is not possible for autoregressive models with more than two lags….

CFA Level 2 – Equity, Session 10 – Reading 35, Equity Valuation: Applications and Processes – LOS d (Practice Questions, Sample Questions) 1. Which of the following is least likely one of the primary steps of the top-down valuation process? A) Selecting a valuation model.B) Assess corporate governance.C) Decision making. Explanation: B) The valuation process consists of…

Slopes Slope (or steepness) of lines is seen everywhere.Engineers refer to the slope of a road as the grade. They often refer to the slope as a percentage.Slope=riserun Slope has no units since it is a ratio. A grade of 8% would mean for every ran of 100 m, there is a rise of 8…

IB Chemistry: First row d-block Elements First Row d-block Elements List the characteristic properties of transition elements Transition metals form : Colored compounds Complex ions Transition metals can exist in Variable Oxidation states Transition Metals can often act as catalysts to reactions In the exam, they might ask you to list a few properties of…

Mean and Variance The main point of this lecture is to tie those concepts back into sigma notation, which is one of the main uses for summation. We’ll walk through this very slowly and unpack it for you. If we have a set of numbers X, with N numbers in them (XI to Xh) they’re…

IGCSE History: World War I – Introduction to the War at Sea Introduction: Britain had not been forced to fight on her own soil. Thus she was not so eager to see Germany pay in the end. Control of the sea was imperative to both Britain and Germany: Both depended on ships to supply them…