Algebra Definitions


absolute comparison	close window
An absolute comparison is an additive comparison between quantities. In an absolute comparison, 7 out of 10 is considered to be larger than 4 out of 5, since 7 is larger than 4.



algorithm	close window
An algorithm is a recipe or a description of a mechanical set of steps for performing some task.



algebraic ideas	close window
Algebraic ideas include patterns, variables, and functions, and are the building blocks of algebraic thinking.



algebraic structure	close window
Algebraic structures are systems with objects and operations, and the rules or properties governing those operations, that can be used to calculate and solve equations.



amplitutde	close window
The amplitude of a cyclic function measures the height of the cyclic function, relative to its average. The amplitude can be determined by the formula: amplitude = (high - low) / 2.



associativity	close window
Associativity is a property of some sets and binary operations where the order in which the operation is carried out may be manipulated to produce the same results. Addition is associative: (2 + 3) + 4 is equivalent to 2 + (3 + 4). Subtraction is not associative: 2 - (3 - 4) is not equivalent to (2 - 3) - 4.



axes	close window
The axes of a graph are the base (or zero) values of two quantities that are being compared in a coordinate graph. The horizontal axis is often referred to as the x-axis and the vertical axis is often referred to as the y-axis.


backtracking	close window
Backtracking is a method of solving equations that involves undoing the operations in an equation to work backward from an output to an input.



base	close window
In the exponential equation 8 = 2³, 2 is the base and 3 is the exponent.



binary operation	close window
A binary operation is an operation that combines two objects of one type to form another object of the same type. For example, in the calculation 2 + 3 = 5, + is a binary operation. Addition, subtraction, multiplication, division, and exponentiation are all binary operations, but there are many others at the heart of the study of algebraic structures.


closed-form description	close window
A closed-form description of a pattern tells how to get from any input to its output, without having to know any previous outputs. A rule such as "take the input, triple it, and add two" is a closed-form description of a pattern.



closure	close window
A property of some sets and binary operations, closure means that the output of a binary operation is always within the set of the inputs. Addition is closed for the set of even numbers: 4 + 6 = 10; any two even numbers will add up to an even number. Addition is not closed for the set of odd numbers: 3 + 5 = 8, and 8 is not an odd number.



commutativity	close window
A property of some sets and binary operations means that which the order of the objects is not important. Addition is commutative: 2 + 3 is equivalent to 3 + 2. Subtraction is not commutative: 2 � 3 is not equivalent to 3 � 2.



common difference	close window
A table of common differences is formed by finding the difference between successive outputs. A table of second common differences can be formed by finding the difference between the differences. Common difference can be used to determine if a table of values comes from a linear or quadratic function. See Session 7, Part D for more information.



covering up	close window
Covering up is a method of solving equations by covering up a portion of the equation to yield a simpler, one-step equation. For example, if 6 + 2m = 12, cover up 2m to give the equation 6 + ? = 12. Because 6 + 6 = 12, the ? must be 6, so 2m = 6 and m = 3.



cyclic function	close window
A cyclic function is a function whose outputs repeat in a cycle. A traffic light is an example of a cyclic function. Cyclic functions have important applications in astronomy and the study of wave motion, including sound and light waves.


dependent variable	close window
In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.



distributivity	close window
Distributivity is a property of some sets and two binary operations, such as addition and multiplication. For these operations, a(b + c) = (ab) + (ac).



direct variation	close window
A direct variation is a relationship between inputs and outputs in which the ratio of inputs and outputs is always the same.

equivalence

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is a relationship between two quantities, usually represented by the equals sign "(=)". There are three important properties of equivalence:

1.	Every quantity is equivalent to itself (3 = 3)
2.	Equivalency does not depend on the sides of the equation (if 2 + 3 = 5, then 5 = 2 + 3)
3.	With three quantities a, b, and c, if a = b and b = c, then a = c (if 6 + 9 = 15 and 15 = 5 x 3, then 6 + 9 = 5 x 3)

exponent

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In the exponential equation 8 = 2³, 2 is the base and 3 is the exponent.

exponential function

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In an exponential function the independent variable is an exponent in an equation. Functions like y = 2^x and y = 10(.5)^x are exponential functions. An exponential function has a constant ratio between successive outputs. For example, in y = 2^x, each time x grows by 1, y is multiplied by 2.

exponential decay function

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An exponential decay function is an exponential function that decreases. The function y = (.5)^x is an exponential decay function; as x increases, y decreases. The decay of nuclear waste is an example of exponential decay.

exponential growth function

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An exponential growth function is an exponential function that increases. The function y = 3^x is an exponential growth function; as x increases, y increases. Population growth and investment growing with interest are examples of exponential growth.


factors	close window
The factors of a whole number are the numbers that divide evenly into it. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.



false position	close window
False postion is a method of solving equations by assuming a convenient number is the solution, then adjusting that number to find the actual solution.



field	close window
A set S and two operations form a field if three conditions are met: the set is a group under the first operation with commutativity, the set is a group under the second operation when the identity of the first operation is removed, and distributivity is satisfied. All fields share many basic properties, including the ability to solve linear equations. Examples of fields include the real numbers under addition and multiplication, and the integers mod 7 (see modular arithmetic) under addition and multiplication.



figurate number	close window
A figurate number is a number of dots which form a geometric shape. If you make a square with 5 dots on a side, there will be 25 dots; this makes the number 25 a square number. If you make a triangle with 4 dots on a side, there will be 10 dots; this makes 10 a triangular number. Figurate numbers can be formed from pentagons, hexagons, cubes, pyramids, and other geometric shapes.



function	close window
A function is any relationship between inputs and outputs in which each input leads to exactly one output. It is possible for a function to have more than one input that yields the same output.



function machine	close window
A function machine is a way of visualizing functions and their inputs and outputs.


group	close window
A set S and two operations form a ring if the set is a group under the first operation with commutativity, the set has associativity, closure, and identity under the second operation when the identity of the first operation is removed, and distributivity is satisfied. Examples of rings include the integers under addition and multiplication, and the integers mod 10 under addition and multiplication.


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identity element	close window
The identity element is an element that has no effect on other elements under a binary operation. In addition with the set of real numbers, zero is the identity element: 7 + 0 = 7, 0 + 7 = 7, and for any number a, a + 0 = a. In multiplication, 1 is the identity element: 5 x 1 = 5, 1 x 5 = 5, and for any a, a x 1 = a.



independent variable	close window
In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.



input	close window
Almost all rules and algorithms start with an input and end with an output. For example, if a rule says to add three to a number, the input of 8 leads to the output of 11.



integer	close window
An integer is any number which is a counting number, zero, or the opposite of a counting number. The set of integers is sometimes written as {..., -3, -2, -1, 0, 1, 2, 3, ...}.



intercept	close window
An intercept is an intersection of a graph with one of the axes. An intersection with the horizontal axis is often referred to as an x-intercept, and an intersection with the vertical axis is often referred to as a y-intercept..



inverse	close window
The inverse of an element a in a binary operation is the element b, which produces the identity for that operation. In addition with the set of real numbers, zero is the identity, so the inverse of 3 is (-3). This is true because 3 + (-3) = 0, and zero is the identity. In multiplication with the set of real numbers, 1 is the identity. Therefore, the inverse of 5 is (1/5), because 5 x (1/5) = 1.



inverse proportion	close window
An inverse proportion is a function in which the output changes in a reciprocal relationship to the input. In other words, if the input doubles, the output is cut in half.



inverse variation	close window
An inverse variation is a relationship between two variables in which a change in one variable results in an inverse, or reciprocal, change in the other. If one variable is multiplied by 3, the other is divided by 3. Forms of inverse variation are A x B = k or B = k / A, where k is a constant number and A and B are the two variables.



iterate	close window
To iterate a function means to repeat its algorithm, using the previous output as the next input.


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linear relationship	close window
A constant rate produces a linear relationship between two variables.


mathematical thinking tools	close window
Mathematical thinking tools are analytic habits of mind, like reasoning skills, problem solving skills, and the ability to represent a situation in different forms.



modular arithmetic	close window
In modular arithmetic, the result of a calculation is the remainder when dividing by the modulus. For example, 33 mod 7 would give the remainder when 33 is divided by 7 -- 33 divides evenly into 7 four times, leaving a remainder of 5. Therefore, 33 mod 7 = 5. An alternate form is to write 33 = 5 (mod 7). Modular arithmetic is sometimes referred to as remainder arithmetic, since its calculations are done with remainders.


network	close window
A network of function machines is made when two or more function machines are connected so that the output from one function machine becomes the input for the next.


output	close window
Almost all rules and algorithms start with an input and end with an output. For example, if a rule says to add three to a number, the input of 8 leads to the output of 11.



origin	close window
The origin of a coordinate system is the point (0, 0).


parabola	close window
The graph of a quadratic function is called a parabola.



period	close window
The period of a cyclic function measures how long it takes to complete one cycle of the function. For example, the period of time it takes for Earth to revolve around the sun is one year.



polygon	close window
A polygon is a two-dimensional figure made up of sides of any length. A triangle, rectangle, pentagon, and octagon are all polygons. Note that the sides of a polygon do not all have to be the same length.



prime number	close window
A prime number is a whole number with exactly two factors: 1 and the number itself. For example, the numbers 2, 7, 13, 19, and 31 are prime. The numbers 0, 1, 12, and 100 are not prime. In particular, 1 is not a prime because it does not have exactly two factors (1 is the only factor).



proportional relationship	close window
A proportional relationship is a relationship between inputs to outputs in which the ratio of inputs and outputs is always the same.


quadratic function	close window
A quadratic function is a function in which the independent variable is squared. The function y = x² is the most basic quadratic function. All quadratic functions fit the form y = Ax² + Bx + C, where A, B, and C can be any real number (although A cannot be zero). The graph of a quadratic function is called a parabola.


rate	close window
A rate describes how much one variable changes in relation to another.



recursive description	close window
A recursive description of a pattern tells you how to proceed from one step to the next. For example, a recursive description might be, "Add two to the value of the output each time the input goes up by one." The Fibonacci sequence, where each output is the sum of the two numbers before it, is a recursive description of a pattern.



relative comparison	close window
A relative comparison is a multiplicative, or proportional, comparison between quantities. In a relative comparison, 4 out of 5 is considered to be larger than 7 out of 10, since 4/5 is larger than 7/10.


representation	close window
A representation of a problem may be a table, a graph, an equation, or a written rule (in words or symbols).

set

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A set is a list of items, or elements, placed in brackets, for example, {2, 3, 4} or {red, blue, yellow}. Sets may contain an unlimited number of elements: {0, 1, 2, 3, ...}. The set containing no elements, {}, is called the empty set or null set."

slope

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The slope of a line is often described as the ratio of rise to run. Slope is also the amount that the dependent variable changes for each increase by one in the independent variable. The formula for slope is: slope = (change in y) / (change in x).

solution set

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The solution set to an equation is the set of all values for the variables in the equation that make the equation true. Here are some examples:

•	The equation x + 5 = 9 has the solution set {4}, since x = 4 is the only way to make the equation true.
•	The equation x² = 9 has the solution set {3, -3}. This represents the two different numbers that x can be.
•	The equation x + 3 = x + 4 has the solution set {}. This means that there are no values for x which make the equation true. The set {} is referred to as the null set, or empty set.
•	4) The equation 2x + y = 10 has an infinite number of solutions, including (5, 0), (4, 2), and many other ordered pairs of values. The solution set would be written as {(5, 0), (4, 2), ...}.

system of equations

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A system of equations is a set of two or more equations with a common solution. Systems of equations may have multiple variables and multiple solutions. If each equation of a system is graphed, the solutions will be any points where all the graphs intersect.


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variable

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The term variable can have different meanings:

•	An indeterminate, a symbol used to represent generalized arithmetic (such as 0 x a = 0)
•	An unknown, used to represent a particular number (such as a + 3 = 7)
•	A relationship between quantities (such as a + b = 10)
•	A term used to help in understanding the study of algebraic structures (such as a + b = b + a)


whole number	close window
A whole number is a counting number or zero. The whole numbers can be written as the series 0, 1, 2, 3, ... . Negative numbers and fractions are not included in the set of whole numbers.


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