math vocabulary

What number

X

X increased by a percentage

X+(percentage)x

Twice a number

2x

Real number

Any number without an "i" to mean "imaginary number." or ∞. Examples: −3, −2.2, 0, 2.75, 3.1415926535897932348...(pi again!), and 2⎯⎯

Arrangements

Arrangement questions ask you how many arrangements of something are possible, such as how many different ways 4 letters can be arranged. Here are the steps to help you solve arrangement questions: Step 1. Draw a blank for each position. Step 2. Fill in the # of possibilities to fill each position. Step 3. Multiply.

Functions

A function is a set of ordered pairs where no two of the ordered pairs has the same x-value. In a function, each input (x-value) has exactly one output (y-value).•The domain of a function refers to the x-values •The range of a function refers to the y-values

Rational number

A number that can be expressed as the quotient of two integers and has a finite number of decimal places, or has decimal values that repeat in a pattern

Irrational number

A number that cannot be expressed as the quotient of two integers and has an infinite number of decimal places. Examples: 3.1415926535897932348..., e (2.7182818284590...), etc.

Prime

A number whose only factors are 1 and itself. The numbers 2, 3, 5, 7, 11, 13, 17, ...are prime. To create a list of prime numbers: List all the odd numbers. Eliminate all perfect squares. Eliminate all numbers ending in 5. Then eliminate any number divisible by 3. Finally, eliminate any number divisible by 7. The remaining list should be a complete list of prime numbers. The number 1 is not considered prime, and the number 2 is the only even prime number

Arithmetic Sequences

An arithmetic sequence, in which each new term is obtained by adding a fixed number, d, to the previous term. Here's an example of an arithmetic sequence: 3, 7, 11, 15, 19, 23, ... A question about arithmetic sequences may ask for one of two values: (1) the value of a specified term (for example, What is the 100th term of the series?) or (2) the sum of the first n terms (for example, What is the sum of the first 50 terms?). Here is the formula to determine the value of a specified term in an arithmetic sequence: a = a + (n − 1)d n 1 This formula gives the value of the nth term an for a sequence with a first term a1 and a difference d. A formula for the sum of the first n terms in an arithmetic sequence: ∑n=n(a 1 +a n )/2 This formula gives the partial sum ∑ n of the first n terms of a sequence with a first term a1 and an nth term an. You should review both of these formulas before exam day and know how to use them. An arithmetic sequence is one in which the difference between consecutive terms is the same. For example, 2, 4, 6, 8..., is an arithmetic sequence where 2 is the common difference. In an arithmetic sequence, the nth term can be found using the formula a n =a 1 +(n−1)d, an=a1+n-1d, where d is the common difference. A geometric sequence is one in which the ratio between two terms is constant. For example, 12 , 12, 1, 2, 4, 8..., is a geometric sequence where 2 is the common ratio.

Ellipses

An ellipse in the 2-D coordinate plane looks like an oval. Here's the basic equation for an ellipse: 1=(x−h) 2rd/ a 2rd +(y−k) 2rd/ b 2rd •Center of the ellipse: (h,k) •Length of the horizontal axis (width of the ellipse): 2a •Length of the vertical axis (height of the ellipse): 2b If a > b, the ellipse is wider than it is tall; if b > a, the ellipse is taller than it is wide

finding the least common multiple for a pair of numbers

Break down each number into its prime factors using a factor tree: 39: 3 × 13 54: 2 × 3 × 3 × 3 Then count quantity of digits within each [digit group] in the factorization of each number: 39: [3] × [13] (one "3" and one "13") 54: [2] × [3 × 3 × 3] (one "2" and three "3" Now, we have 4 [digit groups]: a single 3, a single 2, a single 13, and three 3's. We can only choose 1 digit group per factor value, and we want to choose the largest digit group for each factor value 2 × 3 × 3 × 3 × 13 = 702 = LCM of 39 and 54

Quotient

Divide

"Is to" or per

Divide (ratio) and often a proportion

Percent

Divide by 100

Is

Equal

Even/Odd

Even numbers: 2, 4, 6, 8...; odd numbers: 1, 3, 5, 7... The number 0 is considered even

Factorials

Factorials are represented by "!" (the exclamation point). The factorial of any positive number (n) is equal to the product of all positive numbers less than or equal to n. For example, 5! = 1 × 2 × 3 × 4 × 5 = 120. You may also see the expression n!, which means 1 × 2 × 3 × . . . × n. When evaluating factorials, cancel out as many of the common terms as possible, as in the following example: 12!/9! =12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1/9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 The terms 1 × 2 × 3 × . . . × 9 in the numerator and the denominator cancel out, leaving you with an answer of: 10 × 11 × 12 = 1,320. Additionally, it is valuable to remember that 0! = 1, not 0

Translations

If you are asked to translate horizontally, only the x values will change. If you are asked to translate vertically, only the y values will change. A translation to the left or down is a negative unit change. A translation to the right or up is a positive unit change. The number of units of translation is the number you'll add or subtract from the original x or y coordinate value. For example: •"Translate 3 units left" means a change of −3 to the original x coordinate. •"Translate 2 units up" means a change of +2 to the original y coordinate.

Reflections

If you are reflecting across the x-axis (the horizontal axis), the y values will be flipped (for example, 1 to −1 or −3 to 3) while the x values are unchanged. If you are reflecting across the y-axis (the vertical axis), the x values will be flipped and the y values will stay the same. Another way to think about it: The new coordinate will be the same on the axis reflected over, and have the opposite sign on the other axis. Point (x, y) reflected over the x-axis: (x, −y) Point (x, y) reflected over the y-axis: (−x, y)

geometric sequence

In a geometric sequence, each new term is equivalent to the previous term multiplied by a common ratio r. Here's an example of a geometric sequence: 2, 8, 32, 128, 512, ... In this example, r = 4, so the next number in the sequence would be 2,048. With geometric sequences, you can find the nth term using the formula a =a (r) n−1 (power) an=a1(r n-1) (power) n 1 Formula to determine the value of a specified term in a geometric sequence: an = a1 (rn−1) This formula gives the value of the nth term an for a sequence with a first term a1 and a common ratio r. Formula for the sum of the first n terms in a geometric sequence: ∑n=a 1 (1−r n )/1−r (Nth power) This formula gives the partial sum ∑ n of the first n terms of a sequence with a first term a1 and a common ratio r.

Radians

Like degrees, a radian is a unit for measuring angles; when trigonometric functions are graphed, the x-axis is almost always measured in radians instead of degrees. Converting between the two types of measurement is easy. A full circle, or 360°, is an angle of 2π radians; 180° is π radians. Here's the formula for converting degrees and radians: degrees×π/180° =radians Remember: Degrees to Radians Radians to Degrees Multiply by π Multiply by 180 Divide by 180 Divide by π

Less

Minus

Of and product

Multiply

Consecutive numbers

Numbers that are next to each other: 5 and 6, 100 and 101, 9 and 10...you get the picture. Just add 1 to get from the first number to the second. If a question problem says "Two consecutive numbers" and doesn't tell you what they are, label them as "x" and "x + 1." If they ask for "Three consecutive numbers," label them as "x" and "x + 1" and "x + 2."

Integer

Numbers without decimals or fractions. Also known as a whole number.

Consecutive even/odd numbers

Pairs of even numbers and odd numbers that are next to each other (skipping the odd and even numbers, respectively, between them): 2 and 4, 57 and 59, 100 and 102...you get the picture. Just add 2 between each number. If a problem says "Two consecutive even numbers" or "Two consecutive odd numbers" and doesn't tell you what they are, label them as "x" and "x + 2." If it asks you for "three consecutive even/odd numbers," that means "x," "x + 2," and "x + 4".

Sum or more than

Plus

Probability

Probability refers to the likelihood that an event will occur. It is always between 0 and 1; an event that will definitely not occur has a probability of 0, whereas an event that will certainly occur has a probability of 1

Ratio and proportions

Ratio is the relation between two quantities expressed as one divided by the other. A proportion indicates that one ratio is equal to another ratio

The Radian Circle

The Radian Circle shows the position of every angle on the coordinate plane, and gives its measurement in both degrees (0°-360°) and radians (0π-2π). You can also use the Radian Circle to find the sin, cos, and tan of each angle measure.

Algebraic Manipulation

The answers to these questions are not always numbers. The answer choices will have variables as well as numbers in them. Some of these questions ask "What is y in terms of x and z?" For some reason, kids seem to hate these. Actually, it is just a fancy way of saying "solve for y" or "use algebra to get y alone." It's no different from "y = ?" Whichever letter is after the "what is . . ." is the variable that you solve for.

Mean

The following are properties of the mean, median, and mode that could be tested on the mathematics test: •The arithmetic mean is equivalent to the average of a series of numbers. Calculate he average by dividing the sum of all of the numbers in the series by the total count of numbers in the series. Average=Sum/Count of Numbers Numbers Example: A student received scores of 80 percent, 85 percent, and 90 percent on 3 mathematics tests. The average score received by the student on those tests is 80 + 85 + 90 80 + 85 + 90 divided by 3, or 255÷3, 255÷3, which is 85 percent

Composite Functions

The mathematics test may ask you to combine two functions, like f(x) and g(x), into a composite function. Instead of simply using x as your input, you'll input the whole function f(x) everywhere you see x in the g(x) function.To find f(g(x)) (sometimes written f ○ g), enter the function g(x) everywhere that x appears in the f(x) equation

Linear Inequalities with One Variable

The symbols >, <, ≥, and ≤ indicate the presence of an inequality. Linear inequalities with one variable are solved in almost the same manner as linear equations (with an = sign) with one variable: by isolating the variable on one side of the inequality. When solving an inequality, you use the same methods that you use when solving an equation with an equals sign (=). The only difference is that when an inequality is involved, you reverse the direction of the inequality when you multiply (or divide!) both sides of the inequality by a negative number: −x > 5 −x(−1) > 5(−1) x < −5 In a square root involving an inequality, the square root of x2 is |x|, not x. Here is a common error of this sort: x2 > 9 x2⎯⎯⎯⎯√ > 9⎯⎯√ x > 3 Incorrect! This error causes you to miss half of the possible solutions. This is how to avoid the error: x2 > 9 x2⎯⎯⎯⎯√ > 9⎯⎯√ |x| > 3 This indicates two ranges: x > 3 and x < -3

Reciprocal Trigonometric Functions

The three reciprocal trigonometric functions are secant (sec), cosecant (csc), and cotangent (cot). Their values are the reciprocals of the sine, cosine, and tangent trigonometric functions, respectively, as follows: csc=1/sin sec=1/cos cot=1/tan cot=1tan This means that CHO SHA CAO csc=hypotenuse/opposite sec=hypotenuse/adjacent cot=adjacent/opposite

Constant

This word really throws some kids, but it just means a letter in place of specific, fixed number, just like a variable. Sometimes we may know the numerical value of the constant, sometimes we won't.

Probability of a Single Event

To determine the probability of an event: divide the number of outcomes that fit the conditions of an event by the total number of outcomes. In the very simplest terms: Count the number of outcomes you WANT, and divide it by the TOTAL number of outcomes. So, remember: probability = want/total

Probability of Multiple Events

Two specific events are considered independent if the outcome of one event has no effect on the outcome of the other event. For example, if you toss a coin, there is a 1 in 2, or 12 12 chance that it will land on either heads or tails. To find the probability of two or more independent events occurring together, multiply the outcomes of the individual events. An example of the probability of multiple events: The probability that both coin tosses will result in heads is or 1/2 ×1/2 or 1/4 .

properties of real numbers

•If any number n lies between 0 and any positive number x on the number line, then 0 < n < x; in other words, n is greater than 0 but less than x. If n is any number on the number line between 0 and any positive number x, including 0 and x, then 0 ≤ n ≤ x, which means that n is greater than or equal to 0 and less than or equal to x. •If any number n lies between 0 and any negative number x on the number line, then − x < n < 0; in other words, n is greater than − x but less than 0. If n is any number on the number line between 0 and any negative number x, including 0 and − x, then − x ≤ n ≤ 0, which means that n is greater than or equal to − x and less than or equal to 0.

Adding Odd and Even Numbers

•ODD + ODD = EVEN •ODD + EVEN = ODD •EVEN + EVEN = EVEN

Median

•The median is the middle value of a series of numbers when arranged smallest-to-largest. Examples: The median of the data set (2, 4, 6, 8, 10) is 6. To find the median in a data set with an even number of items, find the average of the middle two numbers. In the series (3, 4, 5, 6) the median is 4.5. The median of the data set (5, 7, 9, 9, 11, 12, 12, 15, 15, 15, 15) is 12. (It's the number in the physical middle of the list.)

Mode

•The mode is the number that appears most frequently in a series of numbers. Examples: The mode of the data set (2, 3, 4, 5, 6, 3, 7) is 3, because 3 appears twice in the series and the other numbers each appear only once in the series. The mode of the data set (5, 7, 9, 9, 11, 12, 12, 15, 15, 15, 15) is 15 (it occurs most often, 4 times). Statistical data may be shown in a table or graph. Just identify the necessary information from the question, then use the correct formula on the data.