An algorithm is a precisely defined routine procedure that can be applied and systematically followed through to a conclusion.
Operations are associative if the order in which operations take place does not affect the result.
For example, addition of numbers is associative, since the order in which they are added does not change their sum. The corresponding associative law is:
`(a+b)+c=a+(b+c)` for all numbers `a`, `b` and `c`.
Multiplication is also associative, as the product of the numbers does not vary with the order of their multiplication. The corresponding associative law is:
`(ab)c=a(bc)` for all numbers `a`, `b` and `c`.
Subtraction and division are not associative, as the order of operations changes the value of the expression.
Determine or find; for example, a number, answer, by using mathematical processes; obtain a numerical answer showing the relevant stages in the working; ascertain or determine from given facts, figures or information.
Two intersecting number lines are taken intersecting at right angles at their origins to form the axes of the coordinate system; the plane is divided into four quadrants by these perpendicular axes, called the `x`-axis, horizontal line and the `y`-axis (vertical line); the position of any point in the plane can be represented by an ordered pair of numbers, `x`, `y`. These ordered pairs are called the coordinates of the point. This is called the Cartesian coordinate system; the plane is called the Cartesian plane.
Operations are commutative if the order in which terms are given does not affect the result.
The commutative law for addition is:
`a+b=b+a`, for all numbers `a` and `b`.
For example, 3+5=5+3.
The commutative law for multiplication is: `ab=ba`, for all numbers a and b.
For example, 4×7=7×4.
Subtraction and division are not commutative; for example, 5-3≠3-5 and 12÷4≠4÷12.
Multiplication of numbers is said to be ‘distributive over addition’, because the product of one number with the sum of two others equals the sum of the products of the first number with each of the others.
For example, the product of 3 with (4+5) gives the same result as the sum of 3×4 and 3×5:
3×(4+5)=3×9=27 and 3×4+3×5=12+15=27
This distributive law is expressed algebraically as follows:
`a(b+c)=ab+ac`, for all numbers `a`, `b` and `c`.
Two expressions are said to be equivalent if they are equal in value.
Any of the individual flat surfaces of a solid object.
The integers are the “whole numbers” including those with a negative sign ⋯-3, -2, -1, 0, 1, 2, 3⋯. In Latin, the word integer means “whole.” The set of integers is usually denoted by Z. Integers are basic building blocks in mathematics.
The arithmetic mean, `x` ̅, of a list of numbers is the sum of the data values divided by the number of values in the list.
In everyday language, the arithmetic mean is commonly called the average.
The median is the value in a set of ordered set of data values that divides the data into two parts of equal size. When there is an odd number of data values, the median is the middle value. When there is an even number of data values, the median is the arithmetic mean of the two central values.
The mode is the most frequently occurring value in a data set.
order of operations
The order of performing mathematical operations:
- evaluate brackets or grouping symbols first
- evaluate any powers and roots
- working left to right, evaluate any multiplication and division
- working left to right, evaluate any addition or subtraction, may also be known as BODMAS, BIDMAS, BEDMAS, etc.'
Partitioning means dividing a quantity into parts. In early years curriculum, it commonly refers to the ability to think about numbers as made up of two parts, such as, 10 is 8 and 2. In later years it refers to dividing both continuous and discrete quantities into equal parts.
A picture graph is a statistical graph for organising and displaying categorical data.
Place value refers to the value of a digit as determined by its position in a number, relative to the ones, or units, place. For integers, the ones place is occupied by the rightmost digit in the number. The value of the next column, the first after the decimal point, represents tenths of ones and this continues with the value of each corresponding digit being representative of a value 10 times smaller than the previous.
For example, in the number 2 594.6 the 4 denotes 4 ones, the 9 denotes 90 ones or 9 tens, the 5 denotes 500 ones or 5 hundreds, the 2 denotes 2000 ones or 2 thousands and the 6 denotes `6/10`of a one or 6 tenths.
The likelihood or chance of something; the relative frequency of the occurrence of an event as measured by the ratio of the number of cases or alternatives favourable to the event to the total number of cases or alternatives.
The range is the difference between the largest and smallest observations in a data set.
A particular kind of ratio in which the two quantities are measured in different units; for example, the ratio of distance to time, known as speed, is a rate because distance and time are measured in different units, such as kilometres and hours; the value of the rate depends on the units in which the quantities are expressed.
A comparison of two quantities of the same kind; for example, if a recipe uses 2 cups of milk and 3 cups of flour, the ratio of milk to flour is 2 is to 3. This can also be written with a colon, 2:3, or as a fraction, `(2 )/3`.
Non-terminating decimals may be recurring: that is, contain a pattern of digits that repeats indefinitely after a certain number of places.
To reflect the point `A` in an axis of reflection, a line is drawn at right angles to the axis of reflection and the point `A` is marked at the same distance from the axis of reflection as A, but on the other side.
The point `A`' is called the reflection image of `A`.
A reflection is a transformation that moves each point to its reflection image.
Part of a population; a subset of the population, often randomly selected for the purpose of estimating the value of a characteristic of the population as a whole.
The sample space of a chance experiment is the set of all possible outcomes for that experiment.
Sampling is the selection of a subset of data from a statistical population. Methods of sampling include:
- systematic sampling - sample data is selected from a random starting point and using a fixed periodic interval
- self-selecting sampling - non-probability sampling where individuals volunteer themselves to be part of a sample
- simple random sampling - sample data is chosen at random where each member has an equal probability of being chosen
- stratified sampling - after dividing the population into separate groups or strata, a random sample is then taken from each group or strata in an equivalent proportion to the size of that group or strata in the population.
A sample can be used to estimate the characteristics of the statistical population.
A graduated line, as on a map, representing proportionate size.
Execute a drawing or painting in simple form, giving essential features but not necessarily with detail or accuracy. In mathematics, it means to represent by means of a diagram or graph; the sketch should give a general idea of the required shape or relationship and should include features.
A plane figure `f` has line symmetry in a line `m`, if the image of `f` under the reflection in `m`is `f` itself. The line `m`is called the axis of symmetry.
A plane figure`f` has rotational symmetry about a point `O`, if there is a rotation such that the image of `f` under the rotation is`f` itself.
A terminating decimal is a decimal that contains a finite number of digits.
Shifting a figure in the plane without turning it is called translation. To describe a translation in the plane, it is enough to say how far left or right and how far up or down the figure is moved.
A translation is a transformation that moves each point to its translation image.