From Zero to Engineer

Natural numbers, integers and the order of operations

Place value, the number line, comparing integers, arithmetic operators and the order in which operations are carried out.

This lesson starts the course from the very beginning: what natural numbers are, what the place value of a digit means and how to compare integers on a number line. It ends with the order of operations, the rule that decides whether your answer comes out right.

Natural numbers

0, 1, 2, 3, 4, 5, 6, 7, 8, 9 …

Place value: each digit says how many hundreds, tens and ones the number contains.

200 + 40 + 6 = 246

Number line

8 > 5

3 < 6

Integers

Integers - positive numbers, negative numbers, and zero.

-5 < 3

-2 > -4

-3 > -6

2 > -4

-7 < 12

The numbers −10, 4, 0, −13 belong to the set of integers.

Writing negative numbers

$5 - -3$ incorrect

$5 - (-3)$ correct

$7 \cdot -2$ incorrect

$7 \cdot (-2)$ correct

Arithmetic operators

Arithmetic operators (symbols): + , − , · , :

addition → sum

subtraction → difference

8 + (-3) = 8 - 3 = 5

9 - (-6) = 9 + 6 = 15

(-4) + (-8) = (-4) - 8 = -12

(-14) - (-7) = (-14) + 7 = -7

multiplication → product

division → quotient

(-5) \cdot 3 = -15

12 : (-6) = -2

(-2) \cdot (-8) = 16

(-14) : (-7) = 2

Order of operations

\begin{aligned}34 + 10 : (2 - 3) \cdot 5 &= 34 + 10 : (-1) \cdot 5 \\&= 34 - 10 \cdot 5 \\&= 34 - 50 = -16\end{aligned}

1
$2 - 3 = -1$ brackets first
2
$10 : (-1) = -10$ then division
3
$-10 \cdot 5 = -50$ then multiplication
4
$34 - 50 = -16$ addition and subtraction last

Frequently asked questions

In what order are operations carried out?

Brackets first, then multiplication and division from left to right, and addition and subtraction last. In the lesson's example only this order gives −16; working from left to right would give something else.

Is zero a natural number?

In this course, yes - the sequence of natural numbers starts at zero. Note that some textbooks start natural numbers at one, so check the definition used by any other source.

Why is writing 5 − −3 incorrect?

Because two operation signs stand next to each other and it is unclear where the subtraction ends and the negative number begins. Put the negative number in brackets: 5 − (−3).

Properties of operations: commutative, associative and distributive