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Many students who are preparing for the SAT confuse or do not know definitions. Definitions are critical. They help you understand and differentiate a concept from others. They also help you retain the concept in a simple way. And finally, they help you answer the question on your SAT test. Here are the most common SAT math definitions that you should become familiar with. You will see these words throughout the SAT math test, and you have to know how to use them. #14 Mean, Median, Mode and Range Mean, median, and mode are three kinds of "averages". You will encounter them in your SAT test prep as well as in the actual test. The mean is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. The median is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order from smallest to largest, so you will have to sort your list before looking for the median. The mode is the value that occurs most often. If no number in the list is repeated, then there is no mode for the list. The range of a list a numbers is just the difference between the largest and smallest values. Let's take and example. Find the mean, median, mode, and range for the following list of values: 11, 18, 11, 14, 17, 16, 14, 22, 11 We find the mean by adding the numbers and dividing by the total. So we have: (11 + 18 + 11 + 15 + 17 + 16 + 14 + 22 + 11)/9 = 15. The mean is 15. To find the median, we first have to sort the series. 11, 11, 11, 14, 15, 16, 17, 18, 22 The middle number is 15, so the median is 15. The mode is the number that is repeated more often than any other, so 11 is the mode. The largest number is 22 and the smallest is 11, so the range is 2211 = 11. #59 Integers, Prime, Rational, Irrational and Real Numbers Integers are positive and negative whole numbers, and zero; not fractions or decimals. Integers include 1, 2, 3, . . . and with 0 and their negative counterparts −1, −2, −3, . . . A prime number is an integer that has exactly two distinct factors: itself and 1. All prime numbers are positive; the smallest prime number is 2. Two is also the only even prime number. One is not prime. A rational number is any number that can be written as a fraction: a ratio of two integers. All positive and negative integers, fractions, and decimal numbers, except numbers containing weird radicals (such as square root of 2), pi, or e. All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless nonrepeating digits to the right of the decimal point. This includes all numbers with radicals that can't be simplified, such as square root of 2 (perfect squares with radicals, such as square root of 16, don't count because they can be simplified to integers, such as 4). Also, all numbers containing pi or e. Note that repeating decimals like .3333... are rational (they're equivalent to fractions, such as 1/3). Real numbers are all the numbers on the number line, including the integers, the rational numbers, and everything else, which includes, for example, the irrational numbers such as √ 2 and π. Imaginary Numbers are the square roots of negative numbers, that is, any numbers containing i, which represents square root of 1. #1012 Function, Domain and Range A function is a set of ordered pairs in which no two ordered pairs have the same first coordinates and different second coordinates. The domain is the set of all possible input values (commonly the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined. It is quite common for the domain to be the set of all real numbers since many mathematical functions can accept any input. For example, the domain of the function f(x) = 1/(1 − x) is all real numbers except for x = 1, since if x = 1, the denominator is 0 and the function “blows up”. The domain of f(x) = √ x is all positive real numbers, along with zero. The range is the set of all possible output values (commonly the variable y, or sometimes expressed as f(x)), which result from using a particular function. For example, the range of the set {6, 8, 1, 4} is 8 − 1 = 7. Shortest summary: The domain is a set of all the values that go into a function and the output values are called the range. Domain → Function → Range Example: when the function f(x) = x2 is given the values x = {1,2,3,...} then {1,2,3,...} is the domain and {1,4,9,...} is the range. #13 Periodic function Periodic functions are functions whose values repeat on a regular interval. This regular interval is called the function’s period. In other words, the period of a function is the smallest domain containing a full cycle of the function. In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic. #1418 Arc, Chord, Radius, Secant, Tangent Any connected part of a circle is called an arc. A chord is a line segment whose endpoints lie on the circle, but it does not necessarily pass through the center. Therefore, a diameter is always a chord but a chord is not always a diameter. The diameter would be the longest chord in the circle. The radius of the circle is a line segment from the center of the circle to a point on the circle. The plural of radius is radii. An extended line that intersects the circle at two points is called a secant. A line that touches the circle at only one point is called a tangent. #1922 Ratios, Percentages, Proportions, Rates A ratio compares two quantities. It shows you that when you have this much of something, you will need to have that much of something else. Ratios are typically written in one of three ways – as fractions, as numbers separated by a colon, or as words – , 2 : 3, or 2 to 3. Ratios are the simplest mathematical (statistical) tools that reveal significant relationships hidden in mass of data, and allow meaningful comparisons. Some ratios are expressed as fractions or decimals, and some as percentages. A real world application of ratios is in the building of model toys. Many model toy cars come scaled at 1/12th of life size. What this means is that every measure of the toy car equals 12 measures in real life. A percentage is a ration that compares a quantity to 100. For example, 20 is 80% of 25. A proportion is an equation stating that two fractions are equal. For example, the proportion 5/15 = 7/21 shows that 5/15 and 7/21 are the same number. The are both multiples of 1/3. A rate is a fraction that shows the relationship between two quantities with different units. A rate is different than a ratio, which compares quantities within a certain category, like red marbles to blue marbles or boys to girls. Rates often use the word “per,” as in cents per pound or miles per hour. #23 Exponents
An exponent indicates that a number is being multiplied by itself a certain number of times. The number being multiplied is called the base. The raised digit is the exponent, and it tells you how many times a number is being multiplied by itself. For example, 3^5 has 3 as the base and 5 as the exponent.
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