Chapter 6 · Rational numbers
Signed Quantities and the Meaning of Zero
Learn how positive and negative numbers describe quantities on opposite sides of zero, how to write them for real situations, and what zero means in each one.
A Celsius thermometer measures temperature in units called degrees Celsius, written °C. Two towns are each °C away from zero on this scale. In the first town it is raining. In the second town the puddles have turned to ice. These towns do not have the same temperature. The first is °C above zero. The second is °C below zero.
The number tells you how far each temperature is from zero. It does not tell you which side of zero. In this lesson you will meet numbers that tell you both things at once.
To show which side of zero a number is on, you write a sign in front of it. The sign is either the plus sign, , or the minus sign, .
- A positive number is greater than zero. You may write it with a plus sign, as , or with no sign, as . Both mean the same number.
- A negative number is less than zero. You always write it with a minus sign, as , and you read it as negative five. Without the minus sign it would be read as a positive number, which is a different number.
- Zero is neither positive nor negative. It separates the positive numbers from the negative numbers.
Positive and negative numbers are called signed numbers, because a sign tells them apart. The rainy town is at °C and the icy town is at °C.
The number after the sign can be a whole number, a decimal or a fraction, as the table shows.
| Written | Read as | Kind of number |
|---|---|---|
or | positive five, or five | positive |
negative five | negative | |
negative two and five tenths | negative | |
or | positive three fourths, or three fourths | positive |
zero | neither positive nor negative |
Why do you need two kinds of numbers? Many quantities can go in two opposite directions from a starting point: up or down, warmer or colder, added or taken away, after or before.
That starting point is the zero of the situation. It is also called the reference point, because every other value is measured from it.
One direction from zero is positive, and the opposite direction is negative. People usually agree on which is which. For example, elevation is the height of a place measured from sea level, which is the average level of the ocean's surface. Heights above sea level are positive, so depths below sea level are negative. If a situation does not say which direction is positive, state your choice before you write any numbers.
| Quantity and unit | What zero means | Positive means | Negative means |
|---|---|---|---|
Temperature in degrees Celsius (°C) | the temperature at which water normally freezes | warmer than that | colder than that |
Elevation in meters (m) | exactly at sea level | above sea level | below sea level |
Change in a bank account in dollars | no money added and none taken out | money added (a deposit) | money taken out (a withdrawal) |
Time from a rocket launch in seconds | the moment of launch | after launch | before launch |
Key concept
Two opposite directions from zero
Positive and negative numbers work together to describe quantities that go in opposite directions from a chosen zero.
- The zero is the reference point. The situation gives it its meaning.
- The sign tells which side of zero. Positive is one direction and negative is the opposite direction.
- The number after the sign tells how far from zero, counted in the unit of the situation.
Zero itself has no sign. It is neither positive nor negative.
To represent a real quantity with a positive or negative number, ask three questions in order:
- What is zero? Name the reference point and the unit.
- Which direction is positive? The opposite direction is then negative.
- Which side, and how far? Pick the sign for the side. Then write how far the quantity is from zero.
If the quantity is exactly at the reference point, write with no sign.
A signed number with its unit is called a signed quantity. For example, an elevation of meters below sea level is the signed quantity m, where m is short for meters. Zero is sea level and above is positive, so below is negative. Here are its three parts:
Worked example
A temperature below zero
One winter morning, the temperature in a town is 7 degrees Celsius below zero. Write this temperature as a signed number.
Name the zero and the unit. The quantity is temperature, measured in degrees Celsius. On this scale, zero is the temperature at which water normally freezes.
Decide which direction is positive. Temperatures warmer than zero are positive, so temperatures colder than zero are negative.
Find the side and how far. This morning is colder than zero, so the number is negative and takes a minus sign. It is 7 degrees from zero, so write 7 after the sign.
Answer
The temperature is °C, which you read as negative seven degrees Celsius.
Worked example
Above, at and below sea level
Elevation is measured in meters (m) from sea level. A hawk flies 120 meters above sea level. A swimmer floats at the surface of the sea, exactly at sea level. A submarine is 45.5 meters below sea level. Write each elevation as a positive number, a negative number or zero.
Name the zero and the directions. Zero is sea level and the unit is meters. Above sea level is positive, so below sea level is negative.
The hawk is above sea level, on the positive side, 120 meters from zero. A positive number means the same with or without its plus sign.
The swimmer is exactly at the reference point, not above sea level and not below it. That elevation is zero, and zero has no sign.
The submarine is below sea level, on the negative side, 45.5 meters from zero. Keep the decimal as it is and put the minus sign in front.
Answer
The hawk is at m, the swimmer is at m and the submarine is at m.
Look again at the zero in each example. For the temperature, zero was the temperature at which water normally freezes. For the elevations, zero was sea level. Each zero had a clear meaning that came from the situation.
To explain the meaning of zero, name the reference point in words from the situation. A useful sentence starts with Zero means and then says what is true at that point. For example: Zero means the swimmer is exactly at sea level, not above it and not below it.
The meaning changes from one situation to the next. For time counted from a rocket launch, zero means the moment of launch. For the change in a bank account, zero means no money was added and none was taken out.
Worked example
Changes in a savings account
Leila records how the money in her savings account changes each day, in dollars. On Monday she deposits 25.50 dollars. On Tuesday she withdraws 40 dollars. On Wednesday she makes no deposit and no withdrawal. Write each day's change as a positive number, a negative number or zero, and explain what zero means here.
Name the zero and the directions. The quantity is the change in the account, in dollars. Zero is no change. A deposit adds money to the account, so count deposits as positive. A withdrawal takes money out, which is the opposite direction, so withdrawals are negative.
Monday adds 25.50 dollars. Adding money is the positive direction, so the change is positive.
Tuesday takes out 40 dollars. Taking money out is the negative direction, so the change takes a minus sign.
Wednesday has nothing added and nothing taken out. That is exactly the reference point, so the change is zero.
Answer
The changes are dollars on Monday, dollars on Tuesday and dollars on Wednesday. Zero means the amount of money in the account did not change that day. It does not mean the account is empty.
Worked example
Reading a water-level report
A town stores its water in a lake called a reservoir. Each day it reports the water level in feet, compared with the normal level. Above normal is positive. The report says: Monday , Tuesday , Wednesday . Describe each day's water level in words, and explain what zero means here.
Name the zero and the directions. Zero is the normal water level and the unit is feet. Above normal is positive, so below normal is negative.
Monday has a plus sign, so the level is on the positive side. The number 1.5 tells how far, so the water is 1.5 feet above normal.
Tuesday is zero, which has no sign. The water is on neither side, so it is exactly at the normal level.
Wednesday has a minus sign, so the level is on the negative side. The number three fourths tells how far, so the water is three fourths of a foot below normal.
You can also write Wednesday's level as a decimal. Three fourths equals 0.75, because 3 divided by 4 is 0.75. The minus sign stays, because the water is still below normal and just as far from zero.
Answer
On Monday the water is feet above normal, on Tuesday it is exactly at the normal level, and on Wednesday it is foot below normal. Zero means the water is at its normal level. It does not mean the reservoir is empty.
Takeaway: positive and negative numbers describe quantities that go in opposite directions from zero. To write one, name the zero, decide which direction is positive, then give the side and how far. Whenever you meet a zero, say in words what it stands for in that situation.
Next, you will use positive and negative numbers in word problems and learn to spot mistakes in how signed quantities are represented.