Negative Numbers & Directed Numbers

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are written with a minus sign (\(\minus\)) before the number. For example, \(\minus 3\) is a negative number, meaning three less than zero.

On a number line, negative numbers are placed to the left of zero, while positive numbers are to the right. Zero itself is neither positive nor negative.

The key difference between negative and positive numbers is their position relative to zero. Positive numbers represent quantities greater than zero, while negative numbers represent quantities less than zero.

For instance, if the temperature is \(\minus 5\), it means 5 degrees below zero, whereas $+5$ means 5 degrees above zero.

A number line helps visualise this:

... \(\minus 4, \minus 3, \minus 2, \minus 1, 0, 1, 2, 3, 4\) ...

For example, \(\minus 2\) is less than 1 because it lies to the left of 1 on the number line.

Directed Numbers Basics

Directed numbers are numbers that have a direction indicated by a positive ($+$) or negative (\(\minus\)) sign. They show not just size but also direction relative to zero.

Positive numbers have a $+$ sign or no sign at all (e.g., $7$ or $+7$), indicating a direction to the right or upwards on a number line. Negative numbers have a \(\minus\) sign (e.g., \(\minus 7\)), indicating a direction to the left or downwards.

Directed numbers are used in many real-life contexts:

• Temperature: Temperatures below zero are negative (e.g., \(\minus 3^\circ\mathrm{C}\)), while above zero are positive (e.g., $+10^\circ\mathrm{C}$).
• Elevation: Heights above sea level are positive (e.g., $+200\,\mathrm{m}$), depths below sea level are negative (e.g., \(\minus 50\,\mathrm{m}\)).
• Bank accounts: A positive balance means money in the account; a negative balance means an overdraft.

Understanding direction helps interpret these numbers correctly.

Reading Negative and Directed Numbers

When reading numbers with a negative sign, say “minus” before the number. For example, \(\minus 4\) is read as “minus four”.

Positive numbers can be read with “plus” or just the number itself. For example, $+5$ can be read as “plus five” or simply “five”.

Directed numbers often appear with symbols such as:

• Minus sign (\(\minus\)) for negative numbers
• Plus sign ($+$) for positive numbers (sometimes omitted)
• Degree symbol ($^\circ$) for temperature, e.g., \(\minus 2^\circ\mathrm{C}\)
• Units like metres ($\mathrm{m}$) for elevation, e.g., \(\minus 10\,\mathrm{m}\)

Interpreting directed numbers requires understanding their context. For example, a temperature of \(\minus 5^\circ\mathrm{C}\) means five degrees below freezing, while an elevation of \(\minus 20\,\mathrm{m}\) means 20 metres below sea level.

For example, the number \(\minus 7\) in the context of a bank account means you owe $\text{£}7$, while $+7$ means you have $\text{£}7$ in your account.

Ordering Negative and Directed Numbers

Ordering numbers means arranging them from smallest to largest or vice versa. When ordering negative and positive numbers, remember:

• All negative numbers are less than zero.
• Between two negative numbers, the one with the larger absolute value (ignoring the sign) is smaller. For example, \(\minus 5\) is less than \(\minus 2\) because $5 > 2$.
• Zero is greater than any negative number but less than any positive number.

On a number line, numbers increase in value as you move from left to right.

For example, ordering the numbers \(\minus 3, 2, 0, \minus 1, 4\) from smallest to largest:

\(\minus 3, \minus 1, 0, 2, 4\)

Rules for ordering directed numbers:

• Compare signs: negative numbers are always less than positive numbers.
• If both numbers are positive, the one with the greater value is larger.
• If both numbers are negative, the number with the smaller absolute value is larger (closer to zero).

For instance, between \(\minus 4\) and \(\minus 2\), \(\minus 2\) is greater because it is closer to zero.