When it comes to numbers, most people are familiar with whole numbers like 1, 2, 3 and so on. We also commonly use fractions and decimals, such as 1/2 or 0.5. But there is an entire set of numbers that extend below zero on the number line – the negative numbers.
What are negative numbers?
Negative numbers are real numbers that are less than zero. They indicate a lower value or opposite direction than positive numbers. For example, debts and temperatures below zero are described with negative numbers. The negative sign “-” is used to denote a number as negative. Some examples of negative numbers are -1, -99, -0.42, etc.
Negative numbers can be integers like -1, -2, -3 and so on. They can also be fractions like -1/2 or decimals like -0.42. Just like with positive numbers, negative numbers can be written as whole numbers, fractions or decimals.
Negative numbers are essential in mathematics for representing and understanding concepts like opposition, deficiency, and decrease. Along with zero and positive numbers, negative numbers help describe the full number system and allow calculations and operations that would not be possible otherwise.
Number line with negative numbers
The number line provides a visual representation of negative numbers and how they extend left from zero:
As you can see, the positive numbers are to the right of 0, while the negative numbers extend to the left. Counting up from 0 takes you through the positive numbers, while counting down goes into the negative numbers. This reflects how negative numbers are less than zero.
Negative integers like -1, -2, -3… extend negatively away from zero, just as the positive integers 1, 2, 3… extend positively. The same applies for decimals and fractions – e.g. -0.5 and -1/2 are the negative equivalents of the positive 0.5 and 1/2.
Comparing negative numbers
Negative numbers can be compared and ordered just like positive numbers. The rules for comparing negative numbers are:
- A greater negative number represents a lower value.
- The negative number farther from 0 on the number line is smaller.
For example:
- -3
- -1 > -5 because -1 is closer to 0 than -5 is.
- -0.2 > -12 because -0.2 has a greater value.
A simple way to compare negative numbers is to think about their absolute values – the positive version of the number. The negative number with the larger absolute value is smaller, because it is farther from zero on the number line.
Comparing Negative Numbers Examples
| Number 1 | Number 2 | Greater Number |
|---|---|---|
| -3 | -8 | -8 |
| -0.5 | -2.7 | -0.5 |
| -5 | -5 | Equal |
This table shows some examples of comparing negative numbers. -8 is greater than -3 because it is smaller. -0.5 is greater than -2.7 because it is closer to 0. When the numbers are equal, neither is greater.
Operations with negative numbers
The basic arithmetic operations of addition, subtraction, multiplication and division can be applied to negative numbers as well as positive. There are some important rules and results to understand when working with negative number operations:
- Adding a negative number is the same as subtracting the absolute value of that number.
- Subtracting a negative number is the same as adding the absolute value of that number.
- Multiplying two negative numbers results in a positive number.
- Dividing two negative numbers results in a positive number.
- A negative number divided by a positive number results in a negative quotient.
These rules follow logically from the way negative numbers extend on the number line. Let’s look at some examples:
Adding Negative Numbers
Adding a negative number works the same as subtracting a positive number. For example:
-5 + -3 = -8
This is the same as subtracting 3 from -5. The sum is -8, which makes sense if you imagine starting at -5 on the number line and moving 3 units to the left.
Subtracting Negative Numbers
Subtracting a negative number works the same as adding a positive number. For example:
-7 – (-2) = -5
This is the same as adding 2 to -7. The difference is -5, which is what you get when starting at -7 and moving 2 units to the right.
Multiplying Negative Numbers
Multiplying two negative numbers results in a positive number. This makes sense logically, because multiplying two numbers with the “opposite” attribute results in a “normal” positive number. For example:
(-3) x (-2) = 6
The negatives cancel out, leaving the positive product 6.
Dividing Negative Numbers
Dividing two negative numbers also results in a positive number, because the negatives cancel each other out. For example:
(-12) ÷ (-6) = 2
The quotient is simply the absolute values 12 and 6 divided normally.
Negative Number Divided by Positive
When dividing a negative number by a positive, the quotient is negative. This is because division essentially splits into parts, which will be negative when splitting a negative number. For example:
-22 ÷ 11 = -2
This makes sense if you think about splitting -22 into 11 equal parts of -2.
Real-world uses of negative numbers
Negative numbers have many practical applications in the real world. Here are some common examples of how negative numbers are used:
Temperatures
Temperatures below zero degrees Celsius or Fahrenheit make use of negative numbers. For example, -5°C represents a temperature five degrees below the freezing point of water.
Elevations
Elevations and depths below sea level are described using negative numbers. For example, the Dead Sea in Israel is 417 m below sea level, or -417 m.
Accounting
In accounting, negative numbers represent money owed or lost. This includes amounts like bank overdrafts, financial deficits, losses and debts.
Physics
In physics, negative numbers describe opposing directions and values. For example, negative velocity represents motion in the opposite direction.
Electric Charge
Electric charge comes in positive and negative forms, indicated by the positive or negative charge symbol. Negative charge is carried by electrons.
Key points about negative numbers
- Negative numbers represent values less than zero.
- They extend to the left on the number line, opposite positive numbers.
- Negative numbers can be integers, fractions or decimals.
- They allow quantities like temperature and elevation to be expressed below zero.
- Operations like addition and multiplication have rules that apply when using negative numbers.
- Negative numbers have many real-world applications in science, accounting and more.
Conclusion
Negative numbers provide a way to represent quantities below zero, which are essential in math and many real-world contexts. While operating with negative numbers follows some specific rules, arithmetic with negatives uses the same principles as positives. Understanding negative numbers allows us to conceptualize and work with lower, opposite values on the full number line.