# How to quickly add large numbers

Addition (Latin: addere: to add) is the first of the four basic arithmetic operations (the others are subtraction, multiplication and division). You can imagine the addition as follows: You look at two bags that are filled with something (instead of bags you can also imagine piles, stacks, or the like, what counts is that you imagine the things to be combined into a whole ). The number of these things can be described with natural numbers. The addition of two natural numbers describes how the number of things in one bag changes when you add the contents of the other bag.

The two numbers that are added are called summands, the number of objects in one bag after adding, i.e. the result of the addition, is called the sum. To mark an addition, use the "\ (+ \)" symbol. The following applies: Summand + summand = sum.

Examples: In one egg carton there are \ (3 \) eggs, in another there are \ (5 \) eggs. If you add the eggs from one box to the other, the new box contains a total of \ (3 + 5 = 8 \) eggs. There are \ (25 \) matches in a matchbox, \ (7 \) on the table. When you pour the matchbox on the table, there are \ (25 + 7 = 32 \) matches on the table. You can also add several numbers: For example, if you want to calculate \ (4 + 2 + 8 \), you first add \ (4 + 2 \), which is \ (6 \), and then add \ (8 \), that means \ (4 + 2 + 8 = 6 + 8 = 14 \). The number \ (0 \) has a special meaning: If you add \ (0 \), it means that you get to a number of things Nothing adds, an added zero does not change the sum, so \ (a + 0 = a \) holds for every number a. One speaks therefore of the zero neutral element of addition.

The addition of natural numbers can (like everything that can be done with natural numbers) be defined using the Peanoaxioms. Here, too, the induction axiom plays a decisive role: It is completely sufficient to define addition with \ (0 \) and to define what the addition with the successor of a number should mean, if one knows how the addition with a number is defined . If \ (n \) is a natural number and \ (S (n) \) is its successor, then one can define the addition for a natural number \ (m \) as follows:
\ begin {align *}
m + 0: = m
\ end {align *}
and
\ begin {align *}
m + S (n): = S (m + n)
\ end {align *}
The sign ": =" means "is defined as". With the help of this definition, the addition of each number can be traced back to the addition with the predecessor and at the end to the addition with the 0. E.g. you can use it to "prove" (yes, this is a real mathematical proof!) That 9 + 4 = 13. Proceed as follows:
\ begin {align *}
9 + 4 & = 9 + S (3) \
& = S (9 + 3) \
& = S (9 + S (2)) \
& = S (S (9 + 2)) \
& = S (S (9 + S (1))) \
& = S (S (S (9 + S (0)))) \
& = S (S (S (S (9)))) \
& = S (S (S (10))) \
& = S (S (11)) \
& = S (12) \
&=13
\ end {align *}

There are numerous mathematical laws for adding natural numbers. The most important are the associative and commutative laws. You can find them in the article "Laws of Calculation".

The subtraction (Latin: subtrahere: remove, withdraw) of natural numbers is the opposite of addition, it is the second basic arithmetic function. Colloquially, subtracting is also called minus arithmetic or sometimes also subtracting. If you subtract a number \ (a \) from another number \ (b \), you are looking for a number \ (c \) such that \ (b-a = c \). One then writes: \ (c + a = b \) (\ (c \) Plus \ (a \) results in \ (b \)).

For example, if you want to subtract \ (14 \) and \ (5 \), the result is \ (9 \), since \ (9 + 5 = 14 \). One therefore writes \ (14-5 = 9 \). In this calculation \ (14 \) is called the minuend, \ (5 \) the subtrahend, while \ (9 \) is the difference between the two numbers.

Just as addition describes the addition of things mathematically, subtraction describes the removal of things. There are two standard situations:

-You want to know how much is left over when you take something away: For example, if there are \ (19 \) apples in a basket and you remove \ (4 \) from them, \ (19-4 = 15 \) apples are left over.

- You want to know how much you have taken away: If there are \ (12 \) apples in a basket in which there were \ (17 \) apples, then \ (17-12 = 5 \) apples have been taken away.

The difference between two numbers describes, in a sense, the difference between them: \ (9 \) and \ (14 \) differ by \ (5 \), since \ (14-9 = 5 \). If a valley is \ (400m \) above sea level and a mountain is \ (1500m \) above sea level, then the difference in altitude between mountain and valley is \ (1100m \), since \ (1500-400 = 1100 \) is. This property of the difference, describing the difference between two numbers, can be seen in the fact that a number is sought for the height difference between mountain and valley which, when added to the height of the valley, gives the height of the mountain; but that is exactly what subtraction is.

Unlike the addition, the subtraction cannot always be carried out: For example, if you want to calculate \ (4-19 \), you are looking for a number that, when added with \ (19 \), results in \ (4 \). Since such a number does not exist, the subtraction cannot be performed. Later you will learn that the natural numbers can be expanded in such a way that expressions like \ (4-19 \) will make sense. Such numbers are called negative numbers and, together with the natural numbers, form the whole numbers.

The next chapter then deals with the other two basic arithmetic operations, multiplication and division.

In the chapter "Calculating with place value systems" you will learn how to add / subtract in writing.