A number machine is a way of showing a sequence of operations applied to a number in a fixed order. Number machines help you understand how numbers behave under addition, subtraction, multiplication, and division, and they are especially useful for working forwards and backwards.

 
A number machine takes an input, applies one or more operations, and produces an output.

For example, a machine that adds \(-3\) and then multiplies by \(2\) can be written as:

$$
\text{output} = (x - 3) \times 2
$$

If the input is \(5\):

$$
(5 - 3) \times 2 = 4
$$

Negative numbers follow the same operation rules as positive numbers, but you must take care with signs.

Adding a negative number is the same as subtracting:

$$
7 + (-4) = 3
$$

Subtracting a negative number is the same as adding:

$$
6 - (-5) = 11
$$

For multiplication and division in number machines:

• Same signs give a positive result
• Different signs give a negative result

$$
(-3) \times 4 = -12
$$

$$
(-20) \div (-5) = 4
$$

You may also be asked to work backwards through a number machine to find the original input. This requires using inverse operations in the reverse order.

For example, if a machine multiplies by \(-4\) and then adds \(6\) to give an output of \(-10\):

$$
x \times (-4) + 6 = -10
$$

Undo the steps:

• Subtract \(6\)
• Divide by \(-4\)

$$
-10 - 6 = -16
$$

$$
-16 \div (-4) = 4
$$

So the original input is \(4\).

Always follow the order of operations carefully. When working backwards, reverse both the order and the operation, and keep close track of negative signs, as they have a major effect on the final result.