maths-AQA NUMBERS

  • Created by: cailen1
  • Created on: 24-07-18 16:24

Order of operations

Mathematical operations must be carried out in the correct order. BODMAS and BIDMAS are ways of remembering this order.

BO/IDMAS Brackets Powers/Indices Divide or Multiply - work from left to right

Add or Subtract - work from left to right

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Negative numbers

Negative numbers are numbers that are less than zero.

A number line can be used to order negative and positive numbers.

Inequality symbols

Inequality symbols can be used to show which number is greater. For example, -2 \textless -1 means -2 is less than -1. The wider part of the inequality sign faces the larger number.

-1 \textgreater -2 means -1 is greater than -2 because -1 is further to the right of -2 on the number line.

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Adding and subtracting negative numbers

  • two signs that are different become a negative sign
  • two signs that are the same become a positive sign
  • Examples

    • Same signs give a positive: 3 + (+2) = 3 + 2 = 5
    • Same signs give a positive: 3 - (-2) = 3 + 2 = 5
    • Different signs give a negative: 3 + (-2) = 3 - 2 = 1
    • Different signs give a negative: 3 - (+2) = 3 - 2 = 1
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Multiplying and dividing negative numbers

The rules for multiplying and dividing numbers are:

  • two signs that are different become a negative sign
  • two signs that are the same become a positive sign

    Examples

    • Same signs give a positive: (-4) \times (-5) = 20
    • Same signs give a positive: 20 \div 5 = 4
    • Different signs give a negative: (-14) \div 2 = -7
    • Different signs give a negative: 14 \div -2 = -7
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Converting decimals to percentages

Once a number is written as a decimal, it can be converted to a percentage. Remember that 'per cent' means 'per hundred', so converting from a decimal to a percentage can be done by multiplying by 100 (move the digits two places to the left).

0.375 as a percentage = 0.375 \times 100 = 37.5 \%

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Converting fractions to decimals

To convert a fraction to a decimal, you need to use written division methods to divide the numerator by the denominator.

Example

Convert \frac{3}{8}\: into a decimal.

Divide 3 by 8.

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Key conversions

Key conversions

There are some important fraction/decimal/percentage conversions that you should know.

FractionDecimalPercentage \frac{1}{10} 0.1 10% \frac{1}{5} 0.2 20% \frac{1}{4} 0.25 25% \frac{3}{4} 0.75 75% \frac{1}{2} 0.5 50% \frac{1}{3} 0. \dot{3} 33. \dot{3} \%

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Using fractions

Fractions show parts of whole numbers, for example, the fraction \frac{1}{4} shows a number that is 1 part out of 4, or a quarter.

Fractions are one way of showing numbers that are parts of a whole. Other ways are decimals and percentages. You can also convert between fractions, decimals and percentages. Like whole numbers and decimals, fractions can be either positive or negative. For example, 3 \frac{1}{5} or - \frac{1}{4}.

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Equivalent fractions

Equivalent fractions are fractions that are worth exactly the same even though they are written differently. \frac{1}{4} is worth the same as \frac{2}{8} because \frac{2}{8} will simplify to \frac{1}{4} by dividing the numerator and denominator by 2.

Working out equivalent fractions

Equivalent fractions are made by multiplying or dividing the denominator and numerator of the fraction by the same number.

For example, to find fractions that are equivalent to \frac{1}{3}, multiply the numerator and denominator by the same number.

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Mixed numbers and improper fractions

2\frac{1}{2}is an example of a mixed number. A mixed number has a whole number part and a fraction part.

The same fraction can also be shown as an improper fraction, \frac{5}{2}. This is equivalent to the mixed number, but in this case the number 2\frac{1}{2} has been written as 5 halves. Improper fractions have numerators which are bigger than the denominators.

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Converting mixed numbers to improper fractions

To convert a mixed number into an improper fraction, look at the denominator of the fraction part first. This will be the denominator of the improper fraction.

Example

Convert 3\frac{1}{2} into an improper fraction.

Change 3\frac{1}{2} into halves. 3 whole ones is 3 \times 2 = 6 halves.

There is another half in the fraction part of 3\frac{1}{2}, so altogether there are 7 halves, meaning that 3\frac{1}{2} is the same as \frac{7}{2}.

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Converting improper fractions to mixed numbers

To convert an improper fraction to a mixed number, work out how many whole numbers there are by dividing the numerator by the denominator. Make the remainder the new numerator and leave the denominator as it was.

Example

Convert \frac{7}{5} into a mixed number.

7 \div 5 = 1 (whole one), and remainder 2.

Write \frac{7}{5} as 1 \frac{2}{5}.

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Fraction arithmetic Adding and subtracting fractio

Fractions with the same denominator can be added (or subtracted) by adding (or subtracting) the numerators.

For instance, \frac{2}{9} + \frac{3}{9} = \frac{5}{9} or \frac{6}{11} - \frac{4}{11} = \frac{2}{11}.

If two fractions do not have the same denominator, then find a common denominator by making equivalent fractions.

Example

Work out \frac{4}{7} + \frac{1}{3}.

Work out the common denominator by looking for the lowest common multiple of 7 and 3. This is 21 (7 \times 3 = 21).

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Multiplying and dividing fractions

Multiplying fractions

To multiply two fractions together, multiply the numerators together and multiply the denominators together.

Example 1

Work out \frac{3}{5} \times \frac{2}{3}.

\frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15}

\frac{6}{15} can be simplified to \frac{2}{5} (by dividing the numerator and denominator by 3).

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Fractions of amounts

Unitary method

A unitary method simply means finding out what 1 of something is worth first.

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Multiplying and dividing fractions

Dividing fractions

To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. This means simply that the divide sign is swapped for a multiply sign, and the second fraction is flipped upside down.

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Highest common factor and lowest common multiple

A common factor is a factor that is shared by two or more numbers. For example, a common factor of 8 and 10 is 2, as 2 is a factor of 8, and 2 is also a factor of 10.

The highest common factor (HCF) is found by identifying all common factors of two numbers and selecting the largest one.

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Lowest common multiple

Lowest common multiple

A common multiple is a number that is a shared multiple of two or more numbers. For example, 24 is a common multiple of 8 and 12, as 24 is in the 8 times tables (8 \times 3 = 24) and 24 is in the 12 times tables (12 \times 2 = 24).

The lowest common multiple (LCM) is found by listing multiples of each number and circling any common multiples. The lowest one is the lowest common multiple.

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Using prime factors to find the HCF and LCM

Using prime factors to find the HCF and LCM

Numbers can be broken down into prime factors using prime factor trees. When the prime factors of two numbers are known, they can be compared to calculate HCFs and LCMs. This can be a more efficient method than listing the factors and multiples of large numbers.

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Law of indices – multiplication

Example

Simplifyc^3 \times c^2.

To answer this question, write c^3 and c^2 out in full:

c^3 = c \times c \times c

c^2 = c \times c.

So, c^3 \times c^2 = c \times c \times c \times c \times c

= c^5

This means c^3 \times c^2 can be simplified to c^5.

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Law of indices – division

Example

Simplifyb^5 \div b^3.

b^5 \div b^3 can be written as \frac{b^5}{b^3}

b^5 \div b^3

b^5 = b \times b \times b \times b \times b and b^3 = b \times b \times b

b^5 \div b^3 so \frac{b^5}{b^3} = \frac{b \times b \times b \times b \times b}{b \times b \times b}

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Standard form

Standard form is written in the form of a \times 10^n, where a is a number bigger than or equal to 1 and less than 10.

n can be any positive or negative whole number.

For example 3.1 × 10^{12}

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Converting between ordinary numbers and standard f

To convert a number into standard form, split the number into two parts - a number multiplied by a power of 10.

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Small numbers-Standard form

A negative power does not mean that the number is negative. It means that we have gone from multiplying by 10 to dividing by 10.

Example

0.03 = 3 \times 10^{-2} because the 3 is 2 places away from the units column.

0.000039 = 3.9 \times 10^{-5} because the 3 is 5 places away from the units column.

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Converting from standard form

To convert a number in standard form to an ordinary number, simply do the multiplication.

Examples

1.34 \times 10^3 is 1,340, since 1.34 \times 10 \times 10 \times 10 = 1,340.

4.78 \times 10^{-3} is 0.00478, as 4.78 \times 0.001 = 0.00478.

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Ordering numbers in standard form

Numbers written in standard form can be ordered by first looking at the power of 10, which tells you the size of the numbers. If two or more numbers have the same power of 10, use the first part of the number to decide the order

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Calculating standard form without a calculator-Add

Adding and subtracting

When adding and subtracting standard form numbers, you have to:

  1. convert the numbers from standard form into ordinary numbers
  2. complete the calculation
  3. convert the number back into standard form
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Calculating standard form without a calculator-Mul

When multiplying and dividing you can use index laws:

  1. multiply or divide the first part of the numbers
  2. apply the index laws to the powers of 10
  3. check whether the first part of the number is between 1 and 10
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Surds

Some numbers cannot be written as exact decimals or fractions for example:

\sqrt{3} = 1.732050807568877 ...

\pi = 3.14159 ...

Roots that cannot be written as exact decimals are called surds. Leaving an answer in surd form means the answer is exact.

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Simplifying surds

Surds can be simplified if the number in the surd has a square number as a factor.

Answers left in surd form are exact answers.

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Adding and subtracting surds

Surds with the same numbers under the roots can be added or subtracted

Example

Simplify 5\sqrt{2} - 3\sqrt{2}

5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}

This is similar to collecting like terms in an expression.

4 \sqrt{2} + 3 \sqrt{3} will not simplify because the numbers inside the square roots, are not the same.

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Multiplying and dividing surds

Multiplying surds with the same number inside the square root

We know that:

\sqrt{2} \times \sqrt{2} = 2

\sqrt{5} \times \sqrt{5} = 5

So multiplying surds that have the same number inside the square root gives a whole, rational number.

(\sqrt{3})^2 = \sqrt{3} \times \sqrt{3} = \sqrt{9} = 3

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Multiplying surds with different numbers inside th

First, simplify the numbers inside the square roots if possible, then multiply them.

Multiply

First multiply the whole numbers:

Then multiply the surds

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Dividing surds

Just like the method used to multiply, the quicker way of dividing is by dividing the component parts

Divide the whole numbers

Divide the square roots

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