Math Articles

Cheat Sheet / Updated 02-16-2023

Some of the most important things to remember in AS-level and A-level maths are the rules for differentiating and integrating expressions. This cheat sheet is a handy reference for what happens when you differentiate or integrate powers of x, trigonometric functions, exponentials or logarithms – as well as the rules you need for what to do when they’re combined!

Cheat Sheet / Updated 02-09-2023

Trigonometry is the study of triangles, which contain angles, of course. Get to know some special rules for angles and various other important functions, definitions, and translations. Sines and cosines are two trig functions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’ll want to understand if you plan to study trig for very long.

Cheat Sheet / Updated 01-19-2023

Statistics involves a lot of data analysis, and analysis is built with math notation and formulas — but never fear, your cheat sheet is here to help you organize, understand, and remember the notation and formulas so that when it comes to putting them into practice or to the test, you’re ahead of the game!

Article / Updated 12-21-2022

So, what is an exponent anyway? According to the Oxford dictionary, an exponent is defined as "a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression." Exponents are used in almost all levels of math, from algebra to calculus to physics. Here are two ways you can work with exponents when they show up in formulas and equations. How to multiply exponents You can multiply many exponential expressions together without having to change their form into the big or small numbers they represent. When multiplying exponents, the only requirement is that the bases of the exponential expressions have to be the same. So, you can multiply because the bases are not the same (although the exponents are). To multiply powers of the same base, add the exponents together: If there’s more than one base in an expression with powers, you can combine the numbers with the same bases, find the values, and then write them all together. For example, Here's an example with a number that has no exponent showing: When there’s no exponent showing, such as with y, you assume that the exponent is 1, so in the above example, you write How to divide exponents You can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base. For example, Pretty easy, huh? Now wrap your brain around this: Any number to the power of zero equals 1, as long as the base number is not 0.

Article / Updated 11-04-2022

Listen to the article:Download audio Whether you're leaving a tip at a restaurant or figuring out just how much those stylish shoes are on sale, you can't get away from percentages. While there are numerous percentage calculators online, it's helpful to be able to do some quick math in your head to calculate percentages without any digital assistance. What is percentage? The word percentage comes from the word percent. If you split the word percent into its root words, you see “per” and “cent.” Cent is an old European word with French, Latin, and Italian origins meaning “hundred." So, percent is translated directly to “per hundred.” If you have 87 percent, you literally have 87 per 100. If it snowed 13 times in the last 100 days, it snowed 13 percent of the time. How to find percentage The numbers that you will be converting into percentages can be given to you in two different formats: decimal and fraction. Decimal format is easier to calculate into a percentage. Converting a decimal to a percentage is as simple as multiplying it by 100. To convert .87 to a percent, simply multiply .87 by 100. .87 × 100=87, which gives us 87 percent. Percent is often abbreviated with the % symbol. You can present your answer as 87% or 87 percent — either way is acceptable. If you are given a fraction, convert it to a percentage by dividing the top number by the bottom number. If you are given 13/100, you would divide 13 by 100. 13 ÷ 100 = .13 Then, follow the steps above for converting a decimal to a percent. .13 × 100 = 13, thus giving you 13%. The more difficult task comes when you need to know a percentage when you are given numbers that don’t fit so neatly into 100. Most of the time, you will be given a percentage of a specific number. For example, you may know that 40 percent of your paycheck will go to taxes and you want to find out how much money that is. How to calculate percentage of a specific number This process is the reverse of what you did earlier. First convert the percentage number to a decimal. Then, you divide your percentage by 100. So, 40 percent would be 40 divided by 100. 40 ÷ 100 = .40 Next, once you have the decimal version of your percentage, simply multiply it by the given number (in this case, the amount of your paycheck). If your paycheck is $750, you would multiply 750 by .40. 750 × .40 = 300 Your answer would be 300. You are paying $300 in taxes. Let’s try another example. You need to save 25 percent of your paycheck for the next 6 months to pay for an upcoming vacation. If your paycheck is $1,500, how much should you save? Start by converting 25 percent to a decimal. 25 ÷ 100 = .25 Now, multiply the decimal by the amount of your paycheck, or 1500. 1500 × .25 = 375 This means you need to save $375 from each paycheck.

Article / Updated 10-26-2022

The volume of an object is how much space the object takes up — or, if you were to drop the object into a full tub of water, how much water would overflow. To calculate the volume of a cylinder, you need to know its height and the area of its base. Because a cylinder is a flat-top figure (a solid with two congruent, parallel bases), the base can be either the top or bottom. If you know a cylinder's height and lateral area, but not its radius, you can use the formula for surface area to find the radius, and then calculate the volume from there. The lateral area of a cylinder is basically one rectangle rolled into a tube shape. Think of the lateral area of a cylinder as one rectangular paper towel that rolls exactly once around a paper towel roll. The base of this rectangle (you know, the part of the towel that wraps around the bottom of the roll) is the same as the circumference of the cylinder's base. And the height of the paper towel is the same as the height of the cylinder. Use this formula to calculate the volume of a cylinder Now for a cylinder problem: Here's a diagram to help you. To use the volume formula, you need the cylinder's height (which you know) and the area of its base. To get the area of the base, you need its radius. And to get the radius, you can use the surface area formula and solve for r: Remember that this "rectangle" is rolled around the cylinder and that the "rectangle's" base is the circumference of the cylinder's circular base. You fill in the equation as follows: Now set the equation equal to zero and factor: The radius can't be negative, so it's 5. Now you can finish with the volume formula: That does it.

Article / Updated 10-26-2022

The t-table (for the t-distribution) is different from the z-table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t-table, is a valuable statistics skill. How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t: First, find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n). Next, find the row corresponding to the degrees of freedom (df) for your problem (for example, n – 1). Go across that row to find the two t-values between which your t falls. For example, if your t is 1.60 and your n is 7, you look in the row for df = 7 – 1 = 6. Across that row you find your t lies between t-values 1.44 and 1.94. Then, go to the top of the columns containing the two t-values from Step 2. The right-tail (greater-than) probability for your t-value is somewhere between the two values at the top of these columns. For example, your t = 1.60 is between t-values 1.44 and 1.94 (df = 6); so the right tail probability for your t is between 0.10 (column heading for t = 1.44); and 0.05 (column heading for t = 1.94). The row near the bottom with Z in the df column gives right-tail (greater-than) probabilities from the Z-distribution. Use the t table to find t*-values (critical values) for a confidence interval involving t: Determine the confidence level you need (as a percentage). Determine the sample size (for example, n). Look at the bottom row of the table where the percentages are shown. Find your % confidence level there. Intersect this column with the row representing your degrees of freedom (df). This is the t-value you need for your confidence interval. For example, a 95% confidence interval with df=6 has t*=2.45. (Find 95% on the last line and go up to row 6.) Practice solving problems using the t-table sample questions below For a study involving one population and a sample size of 18 (assuming you have a t-distribution), what row of the t-table will you use to find the right-tail (“greater than”) probability affiliated with the study results? Answer: df = 17 The study involving one population and a sample size of 18 has n – 1 = 18 – 1 = 17 degrees of freedom. For a study involving a paired design with a total of 44 observations, with the results assuming a t-distribution, what row of the table will you use to find the probability affiliated with the study results? Answer: df = 21 A matched-pairs design with 44 total observations has 22 pairs. The degrees of freedom is one less than the number of pairs: n – 1 = 22 – 1 = 21. A t-value of 2.35, from a t-distribution with 14 degrees of freedom, has an upper-tail (“greater than”) probability between which two values on the t-table? Answer: 0.025 and 0.01 Using the t-table, locate the row with 14 degrees of freedom and look for 2.35. However, this exact value doesn’t lie in this row, so look for the values on either side of it: 2.14479 and 2.62449. The upper-tail probabilities appear in the column headings; the column heading for 2.14479 is 0.025, and the column heading for 2.62449 is 0.01. Hence, the upper-tail probability for a t-value of 2.35 must lie between 0.025 and 0.01.

Article / Updated 10-24-2022

The distinction between numbers and numerals is subtle but important. A number is an idea that expresses how much or how many. A numeral is a written symbol that expresses a number. Here are ten ways to represent numbers that differ from the Hindu-Arabic (decimal) system. Tally marks Numbers are abstractions that stand for real things. The first known numbers came into being with the rise of trading and commerce — people needed to keep track of commodities such as animals, harvested crops, or tools. At first, traders used clay or stone tokens to help simplify the job of counting. Over time, tally marks scratched either in bone or on clay took the place of tokens. Bundled tally marks As early humans grew more comfortable letting tally marks stand for real-world objects, the next development in numbers was probably tally marks scratched in bundles of 5 (fingers on one hand), 10 (fingers on both hands), or 20 (fingers and toes). Bundling provided a simple way to count larger numbers more easily. Of course, this system is much easier to read than non-bundled scratches — you can easily multiply or count by fives to get the total. Even today, people keep track of points in games using bundles such as these. Egyptian numerals Ancient Egyptian numerals are among the oldest number systems still in use today. Egyptian numerals use seven symbols. Egyptian Numerals Number Symbol 1 Stroke 10 Yoke 100 Coil of rope 1,000 Lotus 10,000 Finger 100,000 Frog 1,000,000 Man with raised hands Numbers are formed by accumulating enough of the symbols that you need. For example, 7 = 7 strokes 24 = 2 yokes, 4 strokes 1,536 = 1 lotus, 5 coils of rope, 3 yokes, 6 strokes Babylonian numerals Babylonian numerals, which came into being about 4,000 years ago, use two symbols: 1 = Y 10 = < For numbers less than 60, numbers are formed by accumulating enough of the symbols you need. For example, 6 = YYYYYY 34 = << For numbers 60 and beyond, Babylonian numerals use place value based on the number 60. 61 = Y Y (one 60 and one 1) 124 = YY YYYY (two 60s and four 1s) 611 = < (ten 60s and eleven 1s) Ancient Greek numerals Ancient Greek numerals were based on the Greek letters. The numbers from 1 to 999 were formed using the symbols shown: Roman numerals Although Roman numerals are over 2,000 years old, people still use them today, either decoratively (for example, on clocks, cornerstones, and Super Bowl memorabilia) or when numerals distinct from decimal numbers are needed (for example, in outlines). Roman numerals use seven symbols, all of which are capital letters in the Latin alphabet (which pretty much happens to be the English alphabet as well): I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1,000 Mayan numerals Mayan numerals developed in South America during roughly the same period that Roman numerals developed in Europe. Mayan numerals use two symbols: dots and horizontal bars. A bar is equal to 5, and a dot is equal to 1. Numbers from 1 to 19 are formed by accumulating dots and bars. For example, 3 = 3 dots 7 = 2 dots over 1 bar 19 = 4 dots over 3 bars Numbers from 20 to 399 are formed using these same combinations, but raised up to indicate place value. For example, 21 = raised 1 dot, 1 dot (one 20 + one 1) 399 = raised 4 dots over 3 bars, 4 dots over 3 bars (nineteen 20s + three 5s + four 1s) Base-2 (binary) numbers Binary numbers use only two symbols: 0 and 1. This simplicity makes binary numbers useful as the number system that computers use for data storage and computation. Like the decimal system you're most familiar with, binary numbers use place value. Unlike the decimal system, binary place value is based not on powers of ten (1, 10, 100, 1,000, and so forth) but on powers of two (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and so on), as seen here: Binary Place Values 512s 256s 128s 64s 32s 16s 8s 4s 2s 1s Base-16 (hexadecimal) numbers The computer's first language is binary numbers. But in practice, humans find binary numbers of any significant length virtually undecipherable. Hexadecimal numbers, however, are readable to humans and still easily translated into binary numbers, so computer programmers use hexadecimal numbers as a sort of common language when interfacing with computers at the deepest level, the level of hardware and software design. The hexadecimal number system uses all ten digits 0 through 9 from the decimal system. Additionally, it uses six more symbols: A = 10 B = 11 C = 12 D = 13 E = 14 F = 15 Hexadecimal is a place-value system based on powers of 16. Hexadecimal Place Values 1,048,576s 65,536s 4,096s 256s 16s 1s As you can see, each number in the table is exactly 16 times the number to its immediate right. Prime-based numbers One wacky way to represent numbers unlike any of the others is prime-based numbers. Prime-based numbers are similar to decimal, binary, and hexadecimal numbers in that they use place value to determine the value of digits. But unlike these other number systems, prime-based numbers are based not on addition but on multiplication. Prime-Based Place Values 31s 29s 23s 19s 17s 13s 11s 7s 5s 3s 2s You can use the table to find the decimal value of a prime-based number.

Cheat Sheet / Updated 10-10-2022

Here it is. You have this All-in-One reference for concepts and formulas occurring in Algebra II. The material here is grouped by general algebraic content to make it easier to find what you need. The formulas have the standard mathematical format with variables appearing as x, y, and z and the constant numbers appearing as letters at the beginning of the alphabet.

Article / Updated 10-06-2022

Once you have used the rational root theorem to list all the possible rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that when you divide you get a remainder of 0. Once you have a list of possible rational roots, you then pick one and assume that it’s a root. For example, consider the equation f(x) = 2x4 – 9x3 – 21x2 + 88x + 48, which has the following possible rational roots: If x = c is a root, then x – c is a factor. So if you pick x = 2 as your guess for the root, x – 2 should be a factor. You can use long division to test if x – 2 is actually a factor and, therefore, x = 2 is a root. Dividing polynomials to get a specific answer isn’t something you do every day, but the idea of a function or expression that’s written as the quotient of two polynomials is important for pre-calculus. If you divide a polynomial by another and get a remainder of 0, the divisor is a factor, which in turn gives a root. In math lingo, the division algorithm states the following: If f(x) and d(x) are polynomials such that d(x) isn’t equal to 0, and the degree of d(x) isn’t larger than the degree of f(x), there are unique polynomials q(x) and r(x) such that In plain English, the dividend equals the divisor times the quotient plus the remainder. You can always check your results by remembering this information. Remember the mnemonic device Dirty Monkeys Smell Bad when doing long division to check your roots. Make sure all terms in the polynomial are listed in descending order and that every degree is represented. In other words, if x2 is missing, put in a placeholder of 0x2 and then do the division. (This step is just to make the division process easier.) To divide two polynomials, follow these steps: Divide. Divide the leading term of the dividend by the leading term of the divisor. Write this quotient directly above the term you just divided into. Multiply. Multiply the quotient term from Step 1 by the entire divisor. Write this polynomial under the dividend so that like terms are lined up. Subtract. Subtract the whole line you just wrote from the dividend. You can change all the signs and add if it makes you feel more comfortable. This way, you won’t forget signs. Bring down the next term. Do exactly what this says; bring down the next term in the dividend. Repeat Steps 1–4 over and over until the remainder polynomial has a degree that’s less than the dividend’s. The following list explains how to divide 2x4 – 9x3 – 21x2 + 88x + 48 by x – 2. Each step corresponds with the numbered step in the illustration in this figure. The process of long division of polynomials. (Note that using Descartes’s rule of signs, you find that this particular example may have positive roots, so it’s efficient to try a positive number here. If Descartes’s rule of signs had said that no positive roots existed, you wouldn’t test any positives!) Divide. What do you have to multiply x in the divisor by to make it become 2x4 in the dividend? The quotient, 2x3, goes above the 2x4 term. Multiply. Multiply this quotient by the divisor and write it under the dividend. Subtract. Subtract this line from the dividend: (2x4 – 9x3) – (2x4 – 4x3) = –5x3. If you’ve done the job right, the subtraction of the first terms always produces 0. Bring down. Bring down the other terms of the dividend. Divide. What do you have to multiply x by to make it –5x3? Put the answer, –5x2, above the –21x2. Multiply. Multiply the –5x2 times the x – 2 to get –5x3 + 10x2. Write it under the remainder with the degrees lined up. Subtract. You now have (–5x3 – 21x2) – (–5x3 + 10x2) = –31x2. Bring down. The +88x takes its place. Divide. What to multiply by to make x become –31x2? The quotient –31x goes above –21x2. Multiply. The value –31x times (x – 2) is –31x2 + 62x; write it under the remainder. Subtract. You now have (–31x2 + 88x) – (–31x2 + 62x), which is 26x. Bring down. The +48 comes down. Divide. The term 26x divided by x is 26. This answer goes on top. Multiply. The constant 26 multiplied by (x – 2) is 26x – 52. Subtract. You subtract (26x + 48) – (26x – 52) to get 100. Stop. The remainder 100 has a degree that’s less than the divisor of x – 2. Wow . . . now you know why they call it long division. You went through all that to find out that x – 2 isn’t a factor of the polynomial, which means that x = 2 isn’t a root. If you divide by c and the remainder is 0, then the linear expression (x – c) is a factor and that c is a root. A remainder other than 0 implies that (x – c) isn’t a factor and that c isn’t a root.