Numbers help us count, measure, compare, label and calculate. We use them when we tell the time, pay for something, measure a distance, check a score or write a phone number. This lesson explains the main types of numbers in simple language and shows how they are related.
What is a number?
A number is a mathematical idea used to show an amount, a position or a measurement. A numeral is the symbol used to write that number. For example, the number five can be written with the numeral 5 and the word “five”.
A digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. We combine digits to write larger numbers. The number 407, for example, uses the digits 4, 0 and 7.
Natural numbers
Natural numbers are the counting numbers: 1, 2, 3, 4, 5 and so on. We use them when we count objects. If a basket contains six apples, the number 6 tells us how many apples are present. Some books include 0 among the natural numbers, so always check the convention being used.
Whole numbers
Whole numbers are 0, 1, 2, 3, 4, 5 and so on. They contain no negative sign, decimal part or fraction. Every natural number is a whole number, and zero is also a whole number.
Integers
Integers include negative whole numbers, zero and positive whole numbers. Examples are −5, −2, 0, 3 and 11. Integers are useful for temperatures below zero, floors below ground level, losses and movement in opposite directions.
Rational numbers
A rational number can be written as p/q, where p and q are integers and q is not zero. Examples include 3/4, −2/5, 7 and 0.25. An integer is rational because it can be written with denominator 1; for example, 7 = 7/1.
The decimal form of a rational number either ends or repeats. For example, 1/4 = 0.25 and 1/3 = 0.333… .
Irrational numbers
An irrational number cannot be written as a fraction p/q of two integers. Its decimal form does not end and does not repeat in a fixed pattern. Common examples are √2 and π. The value of π begins 3.14159…, but its digits continue without repeating.
Real numbers
Real numbers include all rational and irrational numbers. They are the numbers that can be placed on a number line. Whole numbers, integers, fractions, terminating decimals, repeating decimals, √2 and π are all real numbers.
Even, odd, prime and composite numbers
Even numbers
An even integer is exactly divisible by 2. Its last digit is 0, 2, 4, 6 or 8. Examples are 4, 18, 60 and −12.
Odd numbers
An odd integer is not exactly divisible by 2. Its last digit is 1, 3, 5, 7 or 9. Examples are 7, 25, 91 and −3.
Prime numbers
A prime number is a natural number greater than 1 that has exactly two positive factors: 1 and itself. The first prime numbers are 2, 3, 5, 7, 11 and 13. The number 2 is the only even prime number. The number 1 is not prime because it has only one positive factor.
Composite numbers
A composite number is a natural number greater than 1 that has more than two positive factors. For example, 12 is composite because its factors are 1, 2, 3, 4, 6 and 12. The number 1 is neither prime nor composite.
Place value
The value of a digit depends on its position. In 5,482, the digit 5 means five thousands, 4 means four hundreds, 8 means eight tens and 2 means two ones. Therefore, 5,482 = 5,000 + 400 + 80 + 2. This is called expanded form.
Indian place-value system
In the Indian system, places are grouped as ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores and ten crores. For example, 4,75,23,610 is read as four crore seventy-five lakh twenty-three thousand six hundred ten.
International place-value system
In the international system, places are grouped as ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions and billions. For example, 47,523,610 is read as forty-seven million five hundred twenty-three thousand six hundred ten.
Important number properties
Commutative property
Changing the order does not change a sum or product: 4 + 7 = 7 + 4, and 3 × 5 = 5 × 3. Subtraction and division are not commutative.
Associative property
Changing the grouping does not change a sum or product: (2 + 3) + 4 = 2 + (3 + 4), and (2 × 3) × 4 = 2 × (3 × 4).
Distributive property
Multiplication can be distributed over addition: a × (b + c) = a × b + a × c. For example, 6 × (10 + 2) = 6 × 10 + 6 × 2 = 72.
Identity property
Adding 0 leaves a number unchanged, and multiplying by 1 leaves a number unchanged. Thus, a + 0 = a and a × 1 = a.
Comparing numbers
To compare positive whole numbers, first count their digits. A number with more digits is greater. If they have the same number of digits, compare from the left until the digits differ. For example, 6,482 is greater than 6,398 because the hundreds digit 4 is greater than 3.
Worked examples
Example 1: Classify −8
−8 is an integer because it is a negative whole number. It is also rational because −8 = −8/1, and every rational number is real. It is even because it is divisible by 2.
Example 2: Write 7,305 in expanded form
The digit 7 is in the thousands place, 3 is in the hundreds place, 0 is in the tens place and 5 is in the ones place. Therefore, 7,305 = 7,000 + 300 + 5.
Example 3: Is 29 prime?
The only positive factors of 29 are 1 and 29. Therefore, 29 is a prime number.
Common mistakes
Do not call 1 a prime number. Do not divide by zero, because division by zero is undefined. Remember that a rational number may be a fraction, an integer or a decimal that ends or repeats. Also remember that the minus sign is part of a negative number.
Practice questions
1. Write 9,064 in expanded form. 2. Classify 0 as natural, whole or integer using the convention stated in this lesson. 3. Decide whether 31 is prime or composite. 4. Write 0.75 as a fraction in simplest form. 5. Which is greater: −3 or −8?
Answers
1. 9,000 + 60 + 4. 2. It is a whole number and an integer; it may also be called natural when that convention includes zero. 3. Prime. 4. 3/4. 5. −3 is greater because it lies to the right of −8 on the number line.
Continue learning
Learn related ideas in Fractions, Decimal Numbers, Common Factors and Exponents.