# Category: 整数

## Natural Numbers

Natural numbers are numbers 1,2,3,4,5,..., that are used for counting objects or for indication of serial number of object.

For example, you can count fingers on your hand, count books etc.

Any natural number can be written with the help of digits 0,1,2,3,4,5,6,7,8,9.

## What is Whole Number

Whole numbers are natural numbers together with zero (0).

So, 0 (zero) is a whole number, as well as 1, 2, 3, 4...

The reason why zero is not a natural number is because we can't count with the help of it. Zero, is in some sense nothing, just like when you have no money, it is actually you have zero money.

## Place Value of Whole Numbers

So, what is place value of a whole number?

In a whole number value of digit depends on its place (position) in the number.

The rightmost digit is ones, next to the left tens, next hundreds, then thousands, ten thousands, hundred thousands, millions, ten millions; etc.

## Rounding Whole Numbers

Sometimes we need to round whole numbers to the tens, hundreds etc.

For this we need to know place value of whole numbers.

Let's see how this can be done.

Example 1. Round 236 to the nearest ten.

Find tens in 236: 236. It is 3.

## Whole Numbers on a Number Line

Number line for whole numbers is a picture of horizontal straight line, where each whole number is shown as specially-marked point, evenly spaced on the line.

Although this pictures shows only 6 whole numbers (from 0 to 5), but arrow indicates that number line contains all whole numbers.

## Comparing Whole Numbers

We can compare whole numbers in the following way:

- Greater than. Notation: $$$>$$$
- Less than. Notation: $$$<$$$
- Equal. Notation: $$$=$$$
- Not Equal. Notation: $$$\ne$$$
- Greater than or equal. Notation: $$$\ge$$$
- Less than or equal. Notation: $$$\le$$$

First way to correctly compare whole numbers is use of number line. Remember, that on number line greater number is to the right of smaller number.

## Adding Whole Numbers

Let's start adding whole numbers.

Suppose you have 2 apples, someone gave you 1 apple, how many apples do you have? Probably, you already know, that the answer is 3. This means that 2+1=3.

So, you can think about addition as a process during which you gain something.

## Subtracting Whole Numbers

Subtraction is in some sense inverse of addition.

Let's start subtracting whole numbers.

Suppose you have $20 and your friend $40. Friend gives you $15. You obtain $15 and now you have $20+$15=$35. But you friend "loses" $15 and now has $40-$15=$25.

## Multiplying Whole Numbers

Let's start multiplying whole numbers.

Suppose 5 of your friends gave you 8 apples. How many apples do you have? It is easily to calculate it using addition: 8+8+8+8+8=40.

But it is very long and time consuming to make such calculations. Moreover, suppose, you have 100 friends that give you 5 apples. You need to add five 100 times!

## Multiplication Table

This is the 15 times table.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 9 0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 11 0 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 12 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 13 0 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 14 0 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 15 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225Since it is not matter in which order to multiply numbers (for example $$${8}\times{3}={3}\times{8}={24}$$$), we actually don't need nearly one half of table (ignore blue or pink part of the table). This makes it easier to remember the table.

## Dividing Whole Numbers (Long Division)

In some sense dividing whole numbers is the inverse of multiplying whole numbers.

Result of dividing number $$${a}$$$ by number $$${b}$$$ is number $$${c}=\frac{{a}}{{b}}$$$ such that $$${a}={b}\times{c}$$$.

## Division with Remainder

When we divided whole numbers, we gave such examples that result is whole number.

But in general it is not true.

What if we try to divide 11 by 4?

There are two 4s in 11 and something extra: $$${11}={4}+{4}+{3}$$$ or $$${11}={4}\cdot{2}+{3}$$$.