Terminating Decimals have a fixed number of digits.
You can write a terminating decimal as a a fraction:
1) Find the value of the column containing the last decimal digit.
2) This is the denominator of the fraction.
3) Write the fraction with the numbers before the decimal point as the numerator.
4) Simplify if possible.
EXAMPLE: 0.12 = 12
Recurring D.: 0.2 repeats as 0.222222..., 0.1245 repeats as 0.12451245...
To Convert a recurring decimal into a fraction:
1) Write an equation with the decimal equal to n. n=0.166666...
2) Multiply the decimal by a power of 10. 10n=1.666666...
3) Subtract the smaller from the larger decimal.
4) Divide the answer by the number of n. 5) If the numerator isn't a whole number, multiply top and bottom by a power of 10 so that both are whole
√a x √b = √a x b
Surds are irrational numbers which can be expressed in root form.
Surds cannot be evaluated exactly.
Straight Line Graphs
You only need two points to draw a straight line graph but you should always check with an extra point.
Gradient of a straight line =
Increase in y
Increase in x
Lines with the same gradient are parallel.
The equation of a line can be written in the form y=mx+c, where m is the gradient and c is the y intercept.
To solve simultaneous equations graphically, draw the lines and find the point where they cross.
To solve simultaneous equations by elimination, make the coefficients of either x or y the same, positive and negative, in both equations by multiplying one equation (or occasionally both) by a number, then add or subtract the equations. If the signs are different, add the equations. If the signs are the same, subtract the equations. Then substitute the value of x or y into one of the equations to find the value of the other value.
An Inequality is a mathematical statement involving one or more inequality symbols.
The Inequality symbols are:
< -Less than
≤ -Less than or equal to
> -Greater than
≥ -Greater than or equal to
Angles, Triangles and Quadrilaterals
Angles and Straight Lines
- Angles at a point add up to 360⁰
- Opposite angles are equal
- Angles that make up a half turn add up to 180⁰
Angles and Parallel Lines
- Alternate angles are equal (z angles)
- Corresponding angles are equal (c angles)
Angle at centre is twice angle at circumference
Angles in same segment are equal.
An angle at the circumference which stands on the diameter of the circle is always a right-angle.
Opposite angles in a cyclic quadrilatral add up to 180⁰
A tangent meets the radius of a circle at 90°
Congruency and Similarity
Congruent shapes are identical. They can be placed exactly on top of each other.
Similar shapes have the same shape but are different sizes. For example, all circles are similar.
In similar shapes the corresponding angles are equal and the corresponding sides are in ratio.
Vectors and Vector Geometry
A Vector has both magnitude (size) and direction.
You can use a vector to describe a translation.
When typed, geometric vectors are bold. For example AB = a.
However, when you write them you have to draw a squiggily line underneath. For example AB = a.
A Scatter Graph shows how two sets of data relate to each other.
Correlation is a measure of how the data relates.
The Trend of a time series graph shows the general direction over a period of time.
Seasonal Variations in trend are seen where the patterns of plotted points match to the seasons of the year, for example heating costs (more in winter than summer).
Cyclical Variations are variations where the general shape of the graph has a tendancy to repeat.
Random Variations are unpredictable and can appear in any time series graph.
Time is always shown in the x axis.