Nurturing maths awesomeness
An angle less than a right angle (ie less than 90°). See also obtuse angle, reflex angle and right angle.
A number to addeg in 4 + 3, both 4 and 3 are addends.
Next to.
Where letters are used to represent an unknown number.
An angle is the rotational difference between two lines, normally measured in degrees.
A value close to the actual value of a number
Part of a circle's circumference.
The amount of skin a shape has.
Not symmetrical
One number that represents all the numbers in a set. Average is often used to mean the mean but median and mode are also averages.
One of the scales used to represent a point in a coordinate system. For a 2D graph, there's an x-axis and a y-axis, for a 3D graph there's also a z-axis.
A mnemonic to help you remember which order to carry out operations: Brackets, Indices, Division, Multiplication, Addition and Subtraction.Doing operations in a different order makes the answer wrong. eg 2+1x3=5 but (2+1)x3=9
A mnemonic to help you remember which order to carry out operations: Brackets, Order, Division, Multiplication, Addition and SubtractionDoing operations in a different order makes the answer wrong. eg 2+1x3=5 but (2+1)x3=9
1. The bottom side of a polygon.
2. The plave value scale of a number system. We normally count in base 10 because we have 0-9, and then a 1 in the next column. This is convenient because we have 10 fingers but other number systems exist:
- Binary
- Hexadecimal
- Sexagesimal
A three digit angle measured clockwise from north, eg 030°
Base 2This number system is used in computers because it makes them easy to make. Because you only need 1 and 0, 1 can be electricity, and 0 no electricity. Instead of the columns (from right to left) being 1, 10, (10x10=) 100, (10x10x10=) 1,000 etc, they are 1, 2, 4, 8, etc
So the number 5 in binary would be 1x4, 0x 2, 1x1 = 101.
Now you are ready for my binary joke:
There are 10 types of people in the world, those who understand binary and those who don't
To cut exactly in half
The part of an expression that must be carried out first. eg 2+1x3=5 but (2+1)x3=9If there is a bracket inside another bracket, start with the innermost bracket.
Work out a value. Calculate does not mean you need to use a calculator
A measure of volume. Centi- means a hundredth, so there are a hundred centilitres in a litre. One litre is 1,000 cm3. You can think of this as a shape measuring 10cm x 10cm x 10cm.
A measure of distance. Centi means a hundredth, so there are 100cm in one meter. Things that are 1cm include: the width of a staple, the thickness of a note pad and the diameter of a belly button.
The point around which scaling is defined. If your pencil was scaled by a factor of 3, it would be 3 times as long. If the centre of scaling was 1m to the left, then the scaled pencil would move 2 metres to the right because it is now 3m from the centre of enlargement.
The point that did not move when a shape is rotated. For a wheel the centre of rotation would, of course, be the centre of the wheel.
A straight line between two points on the circumference of a circle.
The perimeter of a circle.
The range of values (ie upper boundary minus lower boundary) represented by a group.
Age range Class width Frequency 4-7 3 4 8-11 3 6 12-15 3 7 15-18 3 5
A number that a variable is multiplied by. eg the coefficient of 3x is 3.
In a table, columns go up and down, and rows go from side to side.
A denominator is common if it's shared between 2 or more fractionsThese two fractions can not be added together because they do not share a common denominator:
12 + 13
So we make a common denominator by multiplying each fraction by the other fraction's denominator:,
36 + 26 = 56
You can place one shape exactly on top of a shape that's congruent with it. You may need to rotate it, translate it and/or reflect it.
A number that is always the same, represented by a symbol or a letter. One example is π (the ratio of a circle's diameter to its perimeter). Another example is c, the speed of light, used in E=mc2.
The normal way of doing things
Add money to a bank account.
The shape when you slice a 3d shape along its length
A 3D shape where every surface is a square.
A number which is the result of multiplying a whole number by itself, twiceeg
4 cubed = 43 = 4 x 4 x 4 = 16 x 4 = 64
A number which is the result of multiplying a whole number by itself, twiceeg
4 cubed = 43 = 4 x 4 x 4 = 16 x 4 = 64
Like a cube but some sides are rectangles rather than squares.
A running total of frequencies. eg
Age range Frequency Cumulative Frequency 4-7 4 4 8-11 6 10 12-15 7 17 15-18 5 22
A time period of 24 hours, normally measured from midnight to midnight. There are 7 days in a week and 365 ¼ in a year (365 years in most years, but 366 in a leap year)
Take money out of a bank account
10 sided polygon
A number that is not a whole number, expressed with a '.' in it, rather than as a fraction. eg 3.14 , 2.7
The number of digits after the decimal point.
To make a quantity smaller
Literally a measure of something. It is commonly used to mean either:
- A measure of temperature, either as degrees centigrade or degrees farenheit
- A measure of an angle. A circle is divided into 360 degrees. Why 360 and not 100? Because 360 has a lot of factors.
The bottom part of a fraction. numeratordenominator or dividenddivisor.See also: common denominator
To use a mathematical function to convey the properties of something.For example:
The green line above is decribed by the function y = - xIt is also described by x = - y but by convention, we write the y to the left of the equals sign.
The straight line length passing through the centre of a circle from one point on a circumference to an opposite point on a circumference.
Subtract the smaller value from the larger value. ie the difference between 5 and 2 is 3.
Showing the time using displayed numbers rather than with hands or a pointer.
How far away something is. eg at its closest, the distance to the moon is 360,000km .You also see distances (in miles) on road signs.
How data is scattered. This can be represented by numbers (ie standard deviation) or as a plot.
A number to divide byeg in 12 ÷ 3, 3 is the dividend.
The bottom part of a fraction. numeratordenominator or dividenddivisor.See also: common denominator
The top part of a fraction. numeratordenominator or dividenddivisor
Used to show that one quantity has the same value as another
A statement showing one expression has the same value as another. eg 2+3=7-2If it has an equals sign in it, it's an equation.
A shape where all sides have the same length.This could be an equilateral triangle, which is also a regular triangle...
...or an irregular equilateral octagon, for example
A regular triangle. Equilateral or equi(equal) lateral(side) means all sides have the same length as each other. In the case of a triangle, if the side lengths are the same, the angles will also be the same as one another.Mathematicians don't like to 'overspecify' things. They (ok, we) like to give a tiny bit of information from which you can imply the rest of what you might want to know.
To find an approximate answer to a problem. This often involves solving the problem with easier numbers. ie 10.2 x 4.8 ≈ 50
A number which is a multiple of 2. All even numbers end in 0, 2 , 4 , 6 or 8. Whole numbers that aren't even are odd. They're not strange; that's just what they're called.
A number which is a multiple of 2. All even numbers end in 0, 2 , 4 , 6 or 8. Whole numbers that aren't even are odd. They're not strange; that's just what they're called.
To multiply out brackets. eg 2(x+3) expanded becomes 2x+6.
Or (x+1)(2x+3) = 2x2+5x+3
Give a reason supporting your answer.
A collection of terms which can contain variables (letters) and numbers, eg 2x+y=8
Extapolation is the act of inferring one or more new data points beyond the end of a data series, based on the assumption that trends present in the data will continue.Compare: interpolation
A number that fits exactly in another number. eg 6 is a factor of 12
The opposte of expand. Take a common factor out of an expression to put it into brackets. eg 2x+6 factorised is s(x+3)
Or 2x2+5x+3 = (x+1)(2x+3)
Numbers. Fifty two in figures (or numerals) is 52.
An equation that describes the relationship between two or more variables.
The number of times something occurs. It's a tool to make data more helpful.For example a dodgeball club may have the following members:
Age range Frequency Cumulative Frequency 4-7 4 4 8-11 6 10 12-15 7 4 17 15-18 5 22
A mesure of how many values a class represents. It is calculated as frequency divided by class width.
One or more mathematical operations carried out on the input. For example, if the function is 2x +1, and the input (x) is 2, the output would be 5 (2 × 2 +1 )
A measure of how steep a line is found by dividing the distance up by the distance across.. You see this expressed as a percentage (ie multiplied by 100) on road signs:
Gram is a measure of mass. 1g is about the weight of a small paperclip or a plastic pen cap. 1 litre of water weighs 1,000g (= 1kg)
Gram is a measure of mass. 1g is about the weight of a small paperclip or a plastic pen cap. 1 litre of water weighs 1,000g (= 1kg)
The Highest Common Factor (HCF) is the largest factor common to a set of numbers. eg the HCF of 12, 24 and 32 is 4
The vertical dimension of a shape or solid.
A heptagon (also known as a septagon) is a 7 sided polygon (shape).
Is a 50p coin a septagon (or heptagon)?
A six sided polygon (shape).A hexagon
Hexagons with friends:
The Highest Common Factor (HCF) is the largest factor common to a set of numbers. eg the HCF of 12, 24 and 32 is 4
Like a bar chart except each bar represents a range of data points. The width of each bar is proportional to the class interval and the area is proportional to the frequency.
A 'flat' orientation. The surface of water is always horizontal.
The longest side of a right angled triangle.
This is essential to know for pythagoras.
The Inter Quartile Range (IQR) is the difference between the upper and lower quartile in a data set.
It gives an indication of how spread out the middle chunk of data is.
An improper fraction is one where the numerator is larger than the denominator.eg
the mixed fraction 1 13 as an improper fraction would be 43
To make a quantity larger.
eg 24 - 4 is the indice or power
Whole number.See also irrational number, negative number, real number.
Coming between two things.
Interpolation is literally inserting something into something else. In maths, we use it to mean the insertion of an intermediate value based on the existing values.Compare:extrapolation
Interpolation is literally inserting something into something else. In maths, we use it to mean the insertion of an intermediate value based on the existing values.Compare:extrapolation
The Inter Quartile Range (IQR) is the difference between the upper and lower quartile in a data set.
It gives an indication of how spread out the middle chunk of data is.
Irrational numbers are never ending and not recurring decimals that can't be written as a fraction of integers.17 is never ending and not recurring but can be written as a fraction so is not irrational.
√2 is irrational.
A polygon that is not regular. ie its sides are not all the same length and/ or its angles are unequal
An iscosceles has two sides the same shape. It is often said of triangles.But can also be said of other shapes:
Give a reason supporting your answer.
The SI unit of mass. About the same mass as a bag of sugar. 1kg is 1,000g
A measure of distance. 1Km=1,000m
In maths, a kite is a quadrilateral with one line of symmetry stretching between two opposite corners. If there is another line of symmetry between the other two corners, it's a rhombus.
The LCM is the smallest multiple common to a set of numbers. The LCM of 6 and 4 is 12. For prime numbers, the LCM will be the product of the numbers.
In a leap year there's an extra day, so there are 366 days in a leap year, instead of 365. Every 4th year is a leap year.The reason for it is to keep summer in the summer months and winter in the winter months.
The longest dimension of an object that is not its height..
A line (or axis) of symmetry is a lline such that what's on one side of the line is mirrored by what's on the other side of the lineA square has 4 lines of symmetry. You can imagine folding it in half along opposite edges, or folding it in half across its corners, and having both sides of the fold exactly matching up.
A line (or axis) of symmetry is a lline such that what's on one side of the line is mirrored by what's on the other side of the lineA square has 4 lines of symmetry. You can imagine folding it in half along opposite edges, or folding it in half across its corners, and having both sides of the fold exactly matching up.
A measure of volume. One litre is 1,000 cm3, so the volume of 10cm x 10cm x 10cm
Loci is the plural of locus. Most formulas you encounter have only one locus. An example of a formula with many loci would be a tangent plot (ie y=tan(x)).
A locus is a set of points which satisfy a condition (such as a formula). Most commonly the points form a line.eg, the green line represents the locus of y = - x.
The lower quartile is the lowest ¼ of data. If asked to find the lower quartile, you are trying to find the data point ¼ of the way along the data.
The smallest value in a data set.
The LCM is the smallest multiple common to a set of numbers. The LCM of 6 and 4 is 12. For prime numbers, the LCM will be the product of the numbers.
A mix of a whole number and a fractioneg
The improper fraction would be 43 is the same as the mixed fraction 1 13
The most common type of average, formed by adding up the data set and dividing by the number of items.
A type of average: the middle value when the data set is sorted form smallest to largest.
A measure of distance. 1km =1,000m, 1,000mm =1m. Some things that are about 1m: the length of a guitar, the width of a door frame, the height of a kitchen counter.
A measure of volume 1cm cubed, or one thousandth of a litre. A teaspoon is about 5ml.
A tiny measure of distance. There are 1,000mm in 1 metre. Things that are 1mm include: a pencil tip, a mustard seed, a sewing needle tip, a grain of sand.
Relating to the mode (average). You could say the mode in a set of human height data is 178cm, or the modal height is 178cm.
The most commonly occurring value in a data set. You can remember it as it's the same word used in fashion a la mode.If there are two (or more) equally common values, there would be two (or more) modes.
A time period dividing the year into 12. There are between 28 and 31 days in a month. You can remember them by the rhyme:30 days hath September, April June and November
All the rest have 31
Except February all alone with 28 or 29 in a leap year.
A number that's been multiplied by another number. eg 6 is a multiple of 2 (and also a multiple of 3)
A number to multiply byeg in 4 × 3, both 4 and 3 are multiplicands.
A positive integer.If you were counting sheep and came up with half a sheep or a negative number of sheep, that would be unnatural. Zero is not a natural number either. A shepherd with zero sheep is not a shepherd.
A number less than zero.The temperature in a normal household freezer is about -20°c. Negative numbers are also seen around money.
Not 1.
A non-unitary fraction is a fraction that has a numerator that is not 1.14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
A 9 sided polygon.
Perpendicular to a tangent of a line.
Numbers. Fifty two in figures (or numerals) is 52.
The top part of a fraction. numeratordenominator or dividenddivisor
The division sign, ÷
An angle between 90° and 180 °.
Angle b is obtuse.
An 8 sided polygon.
A number that is not even. Odd numbers always end in 1, 3, 5, 7, or 9.
A simple function, most commonly addition, subtraction, multiplication or division.
The direction something is facing.
When two or more lines are parallel they are always the same distance apart. In other words the lines are aligned in exactly the same direction.
What would happen if train tracks weren't parallel?
A quadrilateral where each side is parallel to its opposite side. A rectangle is a special parallelogram.
A 5 sided polygon.
The distance round the outside edge of a shape.
Lines which meet at right angles.
The ratio of a circle's circumference to its diameter.By definition: π=circumferencediameter
This ratio stays the same no matter how big or small a circle gets:
Pi is an irational constant. For most uses 3.14 is precise enough.
Pi has been calculated to over 1 trillion digits but such calculations are entirely useless - 39 digits of pi would be enough to calculate the width of the universe to the precision of one atom. NASA use 15 digits.
A shape with straight sides.
A number greater than zero.All natural numbers are positive
eg 24 - 4 is the indice or power
The number of decimal places that an answer can be quoted to.
The number of decimal places that an answer can be quoted to.
A number that has 2 factors - 1 and itself. 1 is not prime because it has only 1 factor. 0 is not prime either.
A 3D shape with the same cross section all the way through it. If you could make it with a cookie cutter, it's a prism.
A measure of how likely an event is (eg rolling a 3 with dice)
The value you get when multiplying two values together. eg The product of 2 and 3 is 6 (=2x3)
A method for finding side lengths in a right angled triangle.
The square on the hypotenuse is equal to the sum of the squares on the other two sides.
Pythagorean triples are sets of whole side lengths that define right angle triangles. These come up often and spotting one saves you having to calculate square roots eg:
3,4,5
5,12,13
8,15,17
7,24,25
20, 21, 29
12, 35, 37
9, 40, 41They can be disguised by scaling. For example a 3,4,5 triangle may be scaled as 6,8,10, or 9,12,15, etc.
An equation where the highest indice is 2. All these equations are quadratic:
- y = x2 + 3x -5
- y = 4x2 + 3x -5
- y = x2 -5
- y = x2 + 3x
A polygon with 4 sides ('quad' is 4 - lateral is 'sides').
A quotient is what is produced by dividing two numbers. This is often encountered as two numbers (ie a fraction or 12)Less commonly a quotient is defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, 7 ÷ 2 has a quotient of 3.
A shape is rotationally symmetrical if it still looks the same after turning it less than 1 complete rotation.An equilateral triangle has 3 degrees of rotational symmetry, one for each place where the shape looks the same (including the full 360°).
Radians are a unit for measuring angles. Instead of dividing a circle into 360 equal parts (like degrees), radians divide a circle into 2 π (6.28) equal parts.Why? Because 1 radian is the same distance as the radius of the circle
Most people prefer degrees but calculators often default to radians.
Plural of radius
The distance from the centre of a circle to its circumference.
A method of reducing a data set's size by choosing random data points. The assumption is that because the sample is chosen at random, that the random sample represents the whole data set.There are lots of examples of this in opinion polls (where people say what they think) and in advertising (8 of 10 women agree their hair is silkier when they use ___). It would not be feasible to ask everyone.
The difference between the largest number and the smallest in a data set.
The factor by which one number varies with another.
A decimal number that could be formed by dividing one whole number by another. You can often spot rational numbers spot because they either end (ie 0.125) or are recurring(eg 0.333333).
One divided by a number. The reciprocal of 2 is ½. The reciprocal of ½ is 2.
A 4 sided shape where each angle is a right angle. A square is a special rectangle where all the sides are the same length.
A decimal number which doesn't end but repeats after the decimal point (eg 1.3333... or 0.090909). It can be shown to be recurring by putting a dot over the recurring number, or all the recurring numbers. (eg. 1.3 or 0.09)
An angle greater than 180°.
A polygon where all sides are equal and all angles are equal.
The way one variable varies with another. For example y=3x+1 or y= 1x
The amount left over when a number can not be divided exactly.eg 7 ÷ 4 = 1 remainder 3
A quadrilateral with two lines of symetry, each from corner to corner.
An angle of 90°
A triangle with a right angle.
Right angled triangles are easy to work with because you can calculate side lengths with pythagoras and angles (and sides too) with simple trigonometry
A triangle with a right angle.
Right angled triangles are easy to work with because you can calculate side lengths with pythagoras and angles (and sides too) with simple trigonometry
To turn a shape a given angle and a given direction. To be precise, a centre of rotation has to be given.
A shape is rotationally symmetrical if it still looks the same after turning it less than 1 complete rotation.An equilateral triangle has 3 degrees of rotational symmetry, one for each place where the shape looks the same (including the full 360°).
A number that is exactly divisible by 10.The more zeros a number has at the end, the rounder it is, eg:
- 872 is not round
- 870 is round
- 900 is rounder than 870
- 1000 is rounder than 900
To make a number easier to deal with but less precise by reducing the amount of significant figures. eg 3.14159 could be rounded as 3.142 or 3.14
In a table, columns go up and down, and rows go from side to side.
A total that is added to with each new number in the data set..eg
Dice throw Running total 2 2 3 5 4 9
The international system of units. It is wrtten SI rather than IS because it was named by the French: Système International.
How many times greater or smaller the length and width or height) a scaled shape will be scaled by.
A Scalene triangle is one where no two sides are equal and no two angles are equal - although either one of these tests enough on its own because if the sides are all different lengths then no two angles can be the same. A right angled triangle can be a scalene but an iscosceles can't.
To change the size of shape. To define it precisely, the Centre of enlargement needs to be deifined.
A line passing through a circle but not through its centre.
A part of the area of a circle formed of two radii and part of the circumference between them.
An area between the circumference of a circle and a chord.
A heptagon (also known as a septagon) is a 7 sided polygon (shape).
Is a 50p coin a septagon (or heptagon)?
A list of numbers which follow a pattern. eg
- 2,4,6,8...
- 50,45,40,35...
- 2,4,8,16,32....
Sexagesimal is literally base 60.It was invented nearly 7,000 years ago and we still use it: 60 seconds make a minute, and then 60 minutes make an hour.
Each of the digits of a number that are used to express it to the required degree of accuracy, starting from the first non-zero digit.
To write an expression in its simplest (lowest) terms. eg:
- 24 = 12
- x2 + 3 + 3x2 + 2x - 7 -3x +2 = 4x2 - x - 2
A closed 3D shape.
Find a missing value. Frequently written as solve for the value to find. eg solve for x.
How fast something is moving, most commonly in miles per hour (ie how many miles something would travel in 1 hour) in the UK.If you can't remember the formula, think of the units. We say 30 miles per hour ie 30mileshour so speed is distance (miles)time (hours)
A special rectangle where all the sides are the same length.
See also: square number.
A number multiplied by itself. It's called square because if you draw what's happening, it's the area of a square with the given side length. Some square numbers include 1 (=1x1) 4 (=2x2), 9 (=3x3), 16(=4x4)
A number to subtracteg in 4 - 3, 3 is the subtrahend.
The answer when adding values together. eg The sum of 3 and 2 is 5.
The total 'skin' area of a 3D shape.
A shape with 1 or more lines of symmetry.
If you can place a mirror on a shape and the reflected portion behind the mirror is the same as the other half, that shape has symmetry. Where you placed the mirror is called a line of symmetry.
You can also think of it as paper shapes - if it's possible to fold the shape in haplf and have the two halves exactly matching up with no overlaps, then the shape is summetrical.
A way of recording a running total, by marking a number as groups of 5 (4 vertical lines and a horizontal one). It is useful for recording a running total.
IIIIII (=7)
There are two uses of the word tangent in mathematics.
- A straight line that touches a point on a curve wihtout passing through it. A tangent to a circle is, like the circumference it touches, perpendicular to the radius at that point.
- The function which maps a gradient on to an angle, shown here with the angle in radians.
Part of an expression.
The international system of units. It is wrtten SI rather than IS because it was named by the French: Système International.
An improper fraction is one where the numerator is larger than the denominator.eg
the mixed fraction 1 13 as an improper fraction would be 43
Reflections, rotations, translations and / or enlargements are all transformations.
To slide a shape along the x or y axis.
A quadrilateral where 2 sides are parallel to one another. The other 2 sides are not parallel to one another, or it would be a parallelogram.
A quadrilateral where 2 sides are parallel to one another. The other 2 sides are not parallel to one another, or it would be a parallelogram.
A list of the outcomes for a series of probable events. To work out the probability of 2 heads in 3 flips of a coin. you would start on the left wiht branches to heads and tails, and then each of those branches would branch again into heads and tails, and those branches would branch further.
A 3 sided polygon.
A series of numbers where the amount added increases by 1 with each number in the series.
A branch of maths concerned with angles and calculations involving angles.
A unitary fraction is a fraction that has a numerator of 1.14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
1Yes, it's that simple. Unitary, just means 1.
A unitary fraction is a fraction that has a numerator of 1.14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
The scale of the quantity that the number expresses. For example, grams, metres, miles per hour, degrees, etc
The upper quartile is the highest ¼ of data. If asked to find the upper quartile, you are trying to find the data point ¾ of the way along the data.
The largest value in a data set.
A numerical quantity.
A letter in an expression. It may represent a fixed number (eg 2x+1=5 can only be true if x =2), or a variable can be a point anywhere along a line (eg y=3x+2)
A point where two straight lines meet.
An up-down orientation..
Where two straight lines meet (forming an X), any pair of opposite angles will be equal.eg A= B and C = D
The horizontal line separating the numerator from denominator in a fraction
The amount of space an object takes up - like a 3D area.
Time period. There are 7 days in a week and roughly 52 weeks in a year.
The side to side distance. eg a swimming pool may be 12m wide (or have a width of 12m).
The side to side distance. eg a swimming pool may be 12m wide (or have a width of 12m).
Where two straight lines meet (forming an X), any pair of opposite angles will be equal.eg A= B and C = D
The horizontal axis. It is sometimes called y=0 because that is the formula of the line that lies along the x-axis.
The x-coordinate where a graph crosses the x-axis (eg y=0)eg, the x intercept of the purple line is -1.
The vertical axis. It is sometimes called x=0 because that is the formula of the line that lies along the y-axis.
The y - coordinate of where a graph crosses the y-axis (eg x=0).eg, the y intercept of the purple line is -2
The amount of time it takes for the Earth to travel once around the sun. A year is 365 days (366 in a leap year), 12 months or roughly 52 weeks.
The third (normally depth) axis for a 3D space or graph.
Where a line crosses two parallel lines, (forming a Z), the internal opposite angles will be equal.In the diagram, where h and h| are parallel, α = β:
A non-unitary fraction is a fraction that has a numerator that is not 1.14 is a unitary fraction.
34 is a non-unitary fraction.
24 is also a non-unitary fraction, even though it can be simplified to a unitary fraction.
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