Arithmetic operations

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Arithmetic operations

Question 1

Calculate the following:

a)

$\var{pos[0]} + \var{pos[1]} + \var{pos[2]}=$ question.score feedback.none Expected answer:

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b)

$\var{pos[3]} +(\var{neg[0]})=$ question.score feedback.none Expected answer:

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c)

$\var{dec[0]} - \var{dec[1]}=$ question.score feedback.none Expected answer:

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d)

$\var{neg[1]} - (\var{neg[2]})=$ question.score feedback.none Expected answer:

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e)

$\var{neg[3]} + \var{dec[2]}=$ question.score feedback.none Expected answer:

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Advice

A sign between numbers usually tells us whether to add or subtract. Sometimes, we can see two signs following each other.

We need to remember that two different signs (+ and - in any order) become a minus (-), while two same signs become a plus (+).

Operation	First sign	Second sign	Resulting sign	Resulting operation	Example
Addition	$+$	$+$	$+$	Addition	$3 + (+2) = 3 + 2 = 5$
Addition	$+$	$-$	$-$	Subtraction	$3 + (-2) = 3 - 2 = 1$
Subtraction	$-$	$+$	$-$	Subtraction	$3 - (+2) = 3 - 2 = 1$
Subtraction	$-$	$-$	$+$	Addition	$3 - (-2) = 3 + 2 = 5$

Another important thing to remember is to press AC on your calculator to clear it before you start the next calculation. If the following calculation starts with a minus, your calculator will assume you want to subtract something from your previous calculation, rather than start a new one.

Question 2

Decide whether each of the following numbers is positive or negative.

a)

$a = (\var{neg[4]}) \times (\var{neg[5]})$

Gap 0

$a$ is positive
$a$ is negative

Expected answer:

$a$ is positive
$a$ is negative

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b)

$b = \var{pos[4]} \times (\var{neg[6]}) \times \var{dec[5]}$

Gap 0

$b$ is positive
$b$ is negative

Expected answer:

$b$ is positive
$b$ is negative

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Advice

The following rules apply to multiplication and division.

First sign	operation	Second sign	Resulting sign	Example
$+$	$\times$ or $\div$	$+$	$+$	$3 \times 2 = 6$
$+$	$\times$ or $\div$	$-$	$-$	$3 \times (-2) = -6$
$-$	$\times$ or $\div$	$+$	$-$	$(-3) \times 2 = -6$
$-$	$\times$ or $\div$	$-$	$+$	$(-3) \times (-2) = 6$

a)

$(\var{neg[4]}) \times (\var{neg[5]})$ will be positive, as negative times negative gives us positive.

b)

$\var{pos[4]} \times (\var{neg[6]}) \times \var{dec[5]}$ will be negative, as positive times negative times positive gives us negative.

To check these:

\[\begin{align} a &=(\var{neg[4]}) \times (\var{neg[5]}) \\&= + \var{neg[4]*neg[5]} \text{ (positive),} \\ b &= \var{pos[4]} \times (\var{neg[6]}) \times \var{dec[5]} \\&= \var{pos[4]*neg[6]*dec[5]} \text{ (negative).} \end{align} \]

Question 3

Wrong order of solving operations can often lead to incorrect answers. Therefore, the order in which we carry out a calculation is important.

BODMAS is a mnemonic which tells us the correct order in which operations should be carried out:

Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction

Apply BODMAS and try to solve these calculations.

Advice

The correct order of carrying out operations can be remembered by the mnemonic BODMAS:

Brackets ⇒ Ordinals ⇒ Division/Multiplication ⇒ Addition/Subtraction

It is important to notice that division and multiplication have the same priority - division does not have a priority over multiplication. Similarly, adition and subtraction also have the same priority. When the order is unclear, we work from left to right.

Note that brackets have the highest priority, but when we evaluate them, we still need to follow BODMAS inside them.

Sometimes, an alternative acronym BIDMAS (Brackets, Indices, ...) is also used.

Division and multiplication have the same priority, so we just work from left to right. $\var{int*int} ÷ \var{int} = \var{int}$ and hence

\[\begin{align} \var{int*int} ÷ \var{int} \times \var{int} &= \var{int} \times \var{int} \\&= \var{int*int} \text{.} \end{align}\]

Similarly, $\var{eint*2} ÷ \var{eint/2} = 4 $ and hence

\[\begin{align} \var{eint*2} ÷ \var{eint/2} \times \var{eint} &= 4 \times \var{eint} \\&= \var{4*eint}\text{.} \end{align}\]

Applying BODMAS, multiplication has a priority over addition. $\var{sint + 2} \times \var{sint} = \var{(sint + 2)*sint}$ and hence

\[\begin{align} \var{sint} + \var{sint + 2} \times \var{sint} &= \var{sint} + \var{(sint + 2)*sint} \\&= \var{sint + (sint + 2)*sint}\text{.} \end{align}\]

Applying BODMAS, multiplication and division have priority over addition and subtraction. $1 \times 0 = 0$ and $\var{bint}\div\var{bint} = 1$ so

\[\begin{align} \var{bint - 15} - 1 \times 0 + \var{bint}\div\var{bint} &= \var{bint - 15} - 0 + 1 \\&= \var{bint - 14}\text{.} \end{align}\]

Roots can be considered as powers, while fractions can be considered as a bracket divided by a bracket.

\[\displaystyle \text{Numerator is considered as a bracket } (\var{oint}^2+ \sqrt{\var{eint*eint}}) \text{ and the denominator as } (3 \times 2 - 2 \times 2)\text{.}\]

Before we evaluate numerator, we calculate powers:

\[\begin{align} \sqrt{\var{eint*eint}} &= \var{eint} \text{,}
\\\var{oint}^2 &= \var{oint*oint} \text{.} \end{align}\]

Before we evaluate denominator we calculate multiplications:

\[\begin{align} 3 \times 2 &= 6 \text{ and } \\ 2 \times 2 &= 4\text{.} \end{align}\]

Performing addition/subtraction as the last step in evaluating numerator/denominator we get:

\[ \begin{align} (\var{oint}^2+ \sqrt{\var{eint*eint}}) &= \var{oint*oint} + \var{eint}
\\&= \var{oint*oint + eint}
\\\text{and}
\\(3 \times 2 - 2 \times 2) &= 6 - 4
\\&= 2 \end{align} \]

So the fraction

\[\begin{align} \displaystyle \frac{(\var{oint}^2+ \var{eint})}{(3 \times 2 - 2 \times 2)} &= \frac{\var{(oint*oint + eint)}}{2}\text{.} \end{align}\]

Evaluating the final bracket we get:

\[(10 - 2) = 8\text{.}\]

As we evaluated all brackets, we can continue with:

\[\displaystyle \frac{\var{oint}^2+ \sqrt{\var{eint*eint}}}{3 \times 2 - 2 \times 2} + (10 - 2) \div \var{pint} = \frac{\var{(oint*oint + eint)}}{2} + 8 \div \var{pint} \]

Now, division has a priority over addition so since $\frac{\var{(oint*oint + eint)}}{2} = \var{(oint*oint + eint)/2}$ and $8 \div \var{pint} = \var{8/pint}$:

\[\begin{align} \frac{\var{(oint*oint + eint)}}{2} + 8 \div \var{pint} &= \var{(oint*oint + eint)/2} + \var{8/pint} \\&= \var{(oint*oint + eint)/2 + 8/pint}\text{.} \end{align}\]

Numbas has failed

Arithmetic operations

Arithmetic operations

Question 1

Question 1

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Advice

Question 2

Question 2

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Gap 0

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Gap 0

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b)

Question 3

Question 3

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