When we first learn to count we use the natural numbers, which are the numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc. These number symbols were developed by people who lived in the area of India 1200 years ago. With natural numbers we can do problems like 7 - 5 and 20 - 3, but we are not able to do problems like 5 - 5.

Cultures many years ago became frustrated because they could not do problems like 5 - 5, so they decided to make a new number, the zero. The zero symbol we use also was developed by the Hindu people in the area of India. Combining the zero with the natural numbers makes a new set of numbers called the whole numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc. The zero makes the set of whole numbers different from the set of natural numbers. Now we can subtract a number from itself, but we still cannot subtract a large number from a small number.

In Hindu writing, "negative" numbers were indicated by placing a dot or a small circle over or beside a number and they used their word for "negative" to describe these numbers. The Chinese used black for "positive" and red for "negative".

But for hundreds of years, people tended to ignore problems where a large number is subtracted from a small number. Around 1500 A.D., mathematicians began to deal formally with negative numbers. In 1572 A.D., the mathematician Bombelli used the "-" symbol in front of a number to show that it was less than zero or negative.

When the negative numbers are put together with the whole numbers, a new set of numbers is formed called the integers. The integers include all the whole numbers and all the negative whole numbers. In the set of integers, the natural numbers are often called positive number, or positive integers. Zero is neither positive nor negative. The set of integers looks like: <-..., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,...->. Notice that the integers extend forever in two, opposite directions: the positive direction and the negative direction.

In everyday world mathematics, there are six different kinds of numbers that we use: natural (counting), whole, integers, rational, irrational, and real numbers. We've already looked at natural, whole and integers. Rational numbers include any number that can be written as a ratio - thus the name rational. When we use fractions and most decimals we are using rational numbers, but integers are also rational numbers. Irrational numbers are all the numbers that are not rational (the prefix "ir" means "not"). Examples of some common irrational numbers that we use in mathematics are pi, square root of 2, and third root of 4. Those are really "funky numbers". The real numbers are the rational numbers and the irrational numbers combined together, so the real numbers is all numbers we use in everyday life. Rational, irrational, and real numbers can be positive or negative or zero.

Integers are used in the real world. Some examples include the area of finance: when there is more money owed than is available, the person is "in debt" and the amount of debt is a negative number. Very cold temperatures, whether measured in Fahrenheit or Celsius, can be negative numbers. In the game of football, the gain in yardage can be negative when the number of yards lost is more than the number of yards gained. These are just a few examples.

As seen above, it is possible to subtract a large number from a small number; we just get a number that is not a whole number or a natural number. In the early elementary grades, students seldom deal with integers, but deal solely with whole numbers and without integers we cannot subtract a large number from a small number. Using the set of integers, it is incorrect to say that we cannot subtract a large number from a small number.