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When we compute the probability of event $F$ assuming that the event $E$ has already occurred, we call this the conditional probability of $F$ given $E$. We denote this probability as $P(F|E)$. We read $P(F|E)$ as “the probability of $F$ given that $E$ has occurred”, or in a quicker way,…
If m is an integer greater than 1, then a modulo-m system consists of the numbers 0,1,2,…,m−1. Counting and arithmetic operations are performed in a manner corresponding to movements on an m-hour clock. The number is called the modulus of the system. Let’s say we wanted to count to 53…
In base-10, place values correspond to powers of 10, for example: In base-b, place values correspond to powers of b: Base-5 Base-5 uses the digits 0, 1, 2, 3, 4 to represent any number. In base-5, we start by counting 1, 2, 3, 4. At 5, however, we must write…
The seventeenth century saw the beginning of the development of instruments to aid calculation, making the process gradually more and more automatic. One of the forefathers of calculating machines were Napier’s bones (also known as Neper’s rods). Neper was the Latin name used for John Napier, 8th Laird of Merchistoun…
The Maya Indians had a numeration system based on the number 20. It used patterns of dots and bars to count, as shown: Q. Convert 8,292 to Mayan notation. Answer:
The Babylonian numeration system is based on powers of 60 (sexagesimal system). There are two symbols: Small numbers are represented much like the Egyptian system. For example: To represent larger numbers, use several groups of symbols, separated by spaces, and multiply the value of these groups by increasing powers of…
The Chinese system uses multiples of powers of 10 such as 10, 100, and 1,000. The system is a multiplicative system: numerals are formed by writing products of integers between 1 and 9 and powers of 10. Q1. Express 二千 in Hindu-Arabic notation. Q2. Express 九百四十二 in Hindu-Arabic notation. Q3. …
The Romans used letters of the alphabet as numerals. Q1. Convert DCLXXVIII to Hindu-Arabic notation. Furthermore, if the value of a numeral is ever less than the value of the numeral to its right, then the value of the left numeral is subtracted from the value of the numeral to…
In the Egyptian numeration system, numerals are formed by combining the symbols that represent various powers of ten: The value of the number is the sum of the values of the numerals. For example, the number 325 is represented as: The order of the symbols is not important, we could…
When to use the fundamental counting principle, permutations formula, and combinations formula.
Proving identities is a major part of any trigonometry course. This involves getting one of the equation to be identical to the other side. Try using one of the following strategies to begin, and then use others if necessary to continue the verification process. First, start with what appears to…
1. Make the angle between the positive $x$-axis and the positive $y$-axis large enough. 2. Break lines. When one line passes behind another, break it to show that it doesn’t touch and that part of it is hidden. 3. Dash or omit hidden portions of lines. Don’t let the line…
How do write the exponential equation $2^x=8$ in logarithmic form? How do you write the logarithmic equation $\log_2 8 = x$ in exponential form? In case you didn’t notice, these two statements are mathematically equivalent, since the solution to both equals $3$, but what just happened? In the exponential equation,…
Since logarithmic functions are the inverse of an exponential functions, they must follow the laws of exponents. The three properties of logs listed below will be used to combine and solve expressions and equations in this section and the next. Each of these properties are derived from the laws of exponents.…
A radical equation is one in which the unknown variable (e.g. $x$) is under the radical sign. For example: $$\sqrt{x+5}=10$$ When it comes to solving radical equations, oftentimes your teacher will expect you to check your answer using the original equation. This involves substituting your answer back into the equation…
Add and Subtract Radicals One reason to learn how to simplify radicals is to be able to combine them through addition and subtraction. Radicals are called similar if they have the same index and same radicand, such as $4\cdot\sqrt[3]{xy}$ and $7\cdot\sqrt[3]{xy}$. Taking the sum of these two terms would result…
A radical is mathematical way to represent fractional exponents. A radical consists of a radical sign, a quantity under the radical sign called the radicand, and the index of the radical. If an expression, let’s say a, is raised to the power of ½, that’s the same as taking the square…
Radicals (√, ∛, ∜, etc.) are an extension of the exponents laws you learned in Part 1 of this course. This section is solely dedicated to the exponent laws. The connection between radicals and exponents is made in the next section, though it’s highly advised that you review these first as they’re easily…
Multiplying Complex Numbers Imaginary and complex numbers are multiplied the same way you multiply polynomials, with the addition of what you learned in the previous lesson about i when raised to varying exponents. Examples are shown below: Similarly, this idea can be expanded to imaginary numbers found within larger algebraic…
All the complex numbers you’ve dealt with this unit have been in rectangular form (a + bi). Polar form complex numbers are written in terms of r (radius length) and an angle (θ). The numbers a and b are related to r and θ by the formulas below: null In the first video…