AQA GCSE Mathematics Number
The unit code is 8300/1H
- Percentages, decimals and fractions
- Converting decimals to fractions
- Recurring/terminating numbers
- Recurring fractions to decimals
- Recurring decimals to fractions
- Rounding Numbers
Definition of Numbers
In Math’s, a set of numbers can be described in different ways. You could have:
Natural numbers: 1, 2, 3, 4, 5…
Positive integers: ⁺1, ⁺2, ⁺3, ⁺4, ⁺5…
Negative integers: ⁻1, ⁻2, ⁻3, ⁻4, ⁻5…
Square numbers: 1, 4, 9, 16, 25, 36…
You also have numbers which are not integers/not whole numbers such as:
Not an integer: 0.5 ½ 2πr -12.32
Rational or Irrational Numbers
A rational number is any number that can be expressed as a faction of two integers.
And an irrational number, you guessed it, is a number which CANNOT be expressed as a fraction , such as π or 1/3
(try putting them into a calculator, the numbers after the decimal will continue onto infinity)
A multiple is a number which can be found on the multiplication table:
So for example:
Multiples of 4 is 4, 8, 12, 16, 20, 24…60, 64, 68, 72 etc…
Multiples of 5 is 5, 10, 15, 20, 25, 30…75, 80, 85, 90 etc…
Multiples of 9 is 9, 18, 27, 36, 45, 54…135, 144, 153 etc…
“The factors of a number are the natural numbers which divide exactly
into that number (without leaving a remainder)”
“The reciprocal of a number is when the index is -1 and the value is given by 1, divided by the base” – This is the official posh way of saying it!
BODMAS: Bracket, Other, Division, Multiplication, Addition, Subtraction
BODMAS is simply an acronym which explains the order of operations to solve an expression. According to BODMAS rule, if an equation contains ①brackets you should first solve or simplify the bracket followed by ② (powers and roots etc.) then ③ division ④ multiplication ⑤addition and subtraction from left to right
Prime numbers are whole numbers greater than 1, that have only two factors – 1 and the number itself.
Prime numbers are divisible only by the number 1 or itself.
For example: 2, 3, 5, 7 and 11 are the first few prime numbers.
A prime factor is a factor which is also prime number. All natural numbers can be written as a product of prime factors.
21 can be written as 3 x 7 where 3 and 7 are prime factors.
60 can be written as 2 x 2 x 3 x 5 where 2, 3 and 5 are prime factors. The prime factors of a number can be found by successively rewriting the number as a product of prime numbers in increasing order (i.e. 2, 3, 5, 7, 11, 13, 17…etc.).
Squared & Cubed Numbers
A square number is a number which is multiplied by itself.
n x n = n 2
Where n is any number
Cubed is a number which is multiplied by itself 3 times.
n x n x n = n3
It is also the number multiplied by a square (but we’ll do more of this in the Algebra section)
n2 x n = n3
The square root of a number, say for example 49 is the number which when squared equals 7.
This is because 7 x 7 = 49
The sign for square root is √. You may also see 2 √, with a 2 at the front. Don’t let this worry you!! It means the same thing!
Cube root similar to above, if you have 27, the cubed root of this number would be 3.
This is because 3 x 3 x 3 = 27
This time, there will be a 3 in front of the square root symbol to signify it is a cubed root, like this 3√
Lowest Common Multiple “(LCM)”
Highest Common Factor “(HCF)”
The HCF is the biggest number that will divide into all the number given
When given two or more numbers and asked to find the HCF, you need to list all the factors (as done below) and find the biggest numbers for both!
Using the example from below, 54.
find the HCF for 36 and 54.
When measuring distance and weight, there are two main systems which are the Imperial System of Measurement or the Metric System of Measurement.
Most countries (including the UK) use the Metric System. Measurements such as meters, grams and kilo are used with the prefixes as kil, milli and centi to count order of magnitude.
Here are a few things to remember that will come in handy day to day:
When multiplying a number by itself you can use the following shorthand.
7 x 7 = 7² you say ‘7 to the power 2’ (or ‘7 squared’).
7 x 7 x 7 = 7³ you say ‘7 to the power 3’ (or ‘7 cubed’).
2³ x 5² = 33 where 2 and 5 are called the bases and the 3 and 2 are called indices.
Laws Of Indices__________
When simplifying calculations involving indices, certain basic rules or laws can be applied, called the laws of indices.
1. When multiplying 2 or more numbers having the same base, the indices are added:
2. When a number is divided by a number having the same base, the indices are subtracted:
3. When a number which is raised to a power is raised to further power, the indices are multiplied.
4. When a number has an index of 0, its value is 1.
5. A number raised to a negative power is the reciprocal of that number raised to a positive power.
6. When a number is raised to a fractional power the denominator of a fraction is the root of the number and the numerator is the power.
When 3 is divided by 4, it may be written as ¾ . This is called a fraction. The number above the line i.e the 3 is called the numerator and the number below the line, i.e 4 is called the denominator.
When the value of the numerator is less than the value of the denominator, the fraction is called a proper fraction; thus ¾ is a proper fraction. When the value of the numerator is greater than the denominator, the fraction is called improper fraction. Thus 7/3 is an improper fraction and can also be expressed as a mixed number, that is, an integer and a proper fraction. Thus the improper fraction 7/3 is equal to the mixed number 2 1/3.
When a fraction is simplified by dividing the numerator and denominator by the same number, the process is called cancelling. Cancelling by 0 is not permissible.
Equivalent/Cancelling Down Fractions
Equivalent fractions are fractions which are equal in value to each other. Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps its value.
Remember the simple rule….
“change the bottom using multiply or divide,
you do the exact same to the bottom”
2. Mixed numbers_______________
When you see 3½, this is a mixed numbers. You have here an integer part and a fraction part. Remember, improper fractions are ones where the top number is larger than the bottom number. You’ll need to be sooooo familiar with these fractions that you can convert between the two. Luckily it’s really easy, once you know how!
3. Multiplying Fractions___________
Multiplying fractions is a piece of cake!!
All you have to do is put the values into fractions, then multiply the numerators and multiply the denominators.
3. Dividing Fractions___________
Dividing fractions is just as easy, simply flip the second fraction then do the multiplying technique you learned earlier.
5. Common Denominators -Preparation For Adding/Subtracting Fraction_________
You will be asked to add and subtract fractions but also order them by size. To do any of this, you must first find the common denominators for all the fractions given.
The simplest way is to find the lowest common multiple of the denominators. (like we did earlier!!)
6. Adding & Subtracting Fraction_______
Just like above, when adding or subtracting a fraction we need to first find the lowest common multiple of the denominator.
Let’s first look at subtracting fractions with different denominators. A little more challenging but still a piece of cake!
7. Finding The Fraction Of Something___
You may be asked to find the fraction of something. Like “find 4/7 of £120” or “John offers you 1/3 of £400 but Nathan offers you 4/5 of £300. Who is offering you the best deal?” It’s really simple!
Percentage, Decimals & Fractions______
You need to remember percentages, decimals and fractions are just different ways to express the same thing! They all describe the proportion (or the amount) of something. Check out the table below, it’s a good idea to familiarise yourself and know them off the top of your head, don’t worry it’s easy to remember!
It’s good to know these but don’t worry if you haven’t. I’m about to teach you how to convert easily between the 3. This you DO need to learn!
“And if you don’t know, now you know…”
Converting Decimals To Fractions______
Like everything, it’s really easy once you know how, the same applies with decimals to fractions.
All you have to do is move the decimal point. Every time you move the decimal to the right, you add a zero. This new number goes on top and the number of 10s goes underneath. The example will explain it all.
- A recurring decimal is when a number repeats to infinity! Look at 1/6 =0.1666….. Remember it doesn’t have to be a single digit. It could also repeat like: 0.145214521452…
- Normally the repeating part is marked with a dot or a bar on top of the number. If the number is only repeated once, put a single dot on it. If the number repeats in a group of numbers, you put a dot on the first and the last.
- Terminating decimal is when the number comes to an end, like 1/4 = 0.25
Freehand, nooooo calculator, lets Rock’n’roll.
Lets look at a simple example:
There are 2 ways to round numbers,
1. Decimal places
2. Significant figure
You’ll need to read the question careful and see what it is asking you. It’s really easy as it’s in the wording.
Significant figure is identical to that for decimal places except that you start counting from the very beginning of the number (instead of just after the decimal point and after leading zeros)
Occasionally it’s a good idea to estimate a calculation rather than work it out exactly. In this situation, round the numbers in the question before performing the calculation. Usually, numbers are rounded to one significant figure. The ‘approximately equal to’ sign, ≈, is used to show that values have been rounded.
Estimating Square Roots________
Estimating square roots is easy, you just need these 2 steps:
1. Find the squared root on either side of the number given
2. Decide which number is closest and make a sensible estimate of the digit after the decimal point.
And make sure you show you’re working out, gain as many marks as possible!
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