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**natural numbers** - positive whole numbers: 1, 2, 3, 4, to infinity

**counting numbers** - zero and positive whole numbers: 0, 1, 2, 3, 4, to infinity

**integers** - positive and negative whole numbers plus zero : negative infinity ... -3, -2, -1, 0, 1, 2, 3, 4, to positive infinity

**real numbers** - all numbers on a number line including decimals, fractions, and surds

**even numbers** - divide by two exactly e.g. 2, 4, 6, 8

**odd numbers** - do not divide by two exactly e.g. 11, 13, 15, 17

**prime numbers** - have only two factors, one and themselves e.g. 2, 3, 5, 7, 11. Note that 1 is not a prime number because it has only one factor

**surds** - the root of a prime number or the product of different primes

**fractions** - a portion of a whole

**square numbers** - a number multiplied by itself produces a square number e.g. 4, 9, 16

**cube numbers** - a number multiplied by itself three times produces a cube number e.g. 8, 27, 64

**factor** - a number that divides exactly into another number e.g. 3 is a factor of 6

**multiple** - a number in the timetable of another number e.g. 12 is a multiple of 6

**highest common factor** - the largest number that divides into two or more numbers exactly e.g. 6 is the HCF of 24 and 30 (HCF)

**lowest common multiple** - the smallest number that two or more numbers divide into exactly e.g. 120 is the LCM of 24 and 30 (LCM)

**vulgar fraction** - expressed as a whole number over another whole number, respectively numerator and denominator, in which the denominator
cannot be zero; also known as a simple or common fraction

**proper fraction** - a vulgar fraction where the numerator is smaller than the denominator

**improper fraction** - a vulgar fraction where the numerator is greater than the denominator; also called a top-heavy fraction

**rational numbers** - a rational number is any number that can be expressed as a fraction p/q where p and q are integers. Integers are rational numbers as
they can be
written over 1

**irrational numbers** - an irrational number cannot be written as a fraction. For example, the square root of 2 is an irrational number because it cannot
be written as a ratio of two integers

Maths makes use of the following symbols and terms

Symbol | Meaning |
---|---|

= | equal to |

≡ | equivalent to |

≠ | not equal to |

< | less than |

> | greater than |

≤ | less than or equal to |

≥ | greater than or equal to |

≈ | approximately equal to |

∑ | the sum of |

α | alpha |

β | beta |

γ | gamma |

θ | theta |

∞ | infinity |

∴ | therefore |

⇒ | therefore |

Sum | the result of adding |

Difference | the result of substracting |

Product | the result of multiplying |

Prefixes are added to the base unit so 9Gm (giga metres) means 9 x 10^{9} m = 9 000 000 000 m

Index | Symbol | Name |
---|---|---|

10^{12} |
T | tera |

10^{9} |
G | giga |

10^{6} |
M | mega |

10^{3} |
k | kilo |

10^{2} |
h | hecto |

10^{1} |
da | deca |

base unit | ||

10^{-1} |
d | deci |

10^{-2} |
c | centi |

10^{-3} |
m | milli |

10^{-6} |
µ | micro |

10^{-9} |
n | nano |

10^{-12} |
p | pico |

Title | Rule | Examples |
---|---|---|

Divisible by 2 | End with 0,2,4,6,8 | 254 and 4718 are divisible by 2 |

Divisible by 3 | Sum of digits is divisible by 3 | 549 =5 + 4 + 9 = 18 = 1 + 8 = 9 is divisible by 3 hence 549 is divisible by 3 |

Divisible by 4 | Last two digits divisible by 4 | 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is divisible by 4 |

Divisible by 5 | Ends with 0 or 5 | 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5 |

Divisible by 6 | Divides by both 2 and 3 | 4536 here last digit is 6 so it divisible by 2 and 4+5+3+6=18=1+8=9 which is divisible by 3. Hence 4536 is divisible by 6 |

Divisible by 8 | Last 3 digits divide by 8 | 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8 |

Divisible by 10 | End with 0 | 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10 |

Term | Explanation | Illustration |
---|---|---|

semi-circle | half a circle | |

concentric circles | circles which share the same centre | |

circumference | the perimeter of a circle | |

arc | a portion of the circumference | |

minor arc | an arc that is less than half the circumference | |

major arc | an arc that is more than half the circumference | |

radius | the line from the centre of the cirle to the edge of the circle | |

diameter | the line from one side of the circle to the other through the centre | |

chord | a line from one side of the circle to the other side whose endpoints are on the edge of a circle | |

secant | a line that pass through both sides of the circle | |

tangent | a line that touches the circle at one point only | |

normal | a line at 90 degrees to the tangent at the point where it touches the circle | |

segment | the portion of a circle between a chord and an arc | |

sector | the portion of a circle between two radii and an arc | |

subtended angle | ||

bisector | a line that cuts another line in half |

Number | Theorem | Diagram |
---|---|---|

Theorem 1 | The angles on the cirumference of a circle are equal when subtended by the same arc | |

Theorem 2 | The angle at the centre of a circle, is twice the size of the angle at the circumference when subtended by the same arc | |

Theorem 3 | An angle on the circumference is 90 degrees when subtended by diameter of the circle | |

Theorem 4 | The opposite angles in a cycle quadrilateral always add up to 180 degrees. | |

Theorem 5 | A tangent to a circle is perpendicular to the radius at the point of contact | |

Theorem 6 | Tangents to a circle from the point they intersect to the points of contact are equal in length | |

Theorem 7 | The line joining the point where two tangents intersect to the centre of the circle, bisects the angle between the tangents | |

Theorem 8 | If a radius bisects a chord, the angle between them will be at 90 degrees | |

Theorem 9 | The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment |

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