Natural numbers II

Go down deep enough into anything and you will find mathematics, Dean Schlicter.

Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted, Albert Einstein.

Basic Arithmetic: Addition I. What is it? How does it work?

The four basic mathematical operations are: addition (+), subtraction(-), multiplication(x), and division(:). To understand addition have a look at this picture.

There are two sets or group of apples. Let’s count them. There are three apples on the left and two on the right. How many apples do I have in total now? You’ve guessed it! We have five apples. In other words, three plus two equals five: 3 + 2 = 5. Three and two are the addends of the addition problem and five is the result of the sum.

Adding two numbers together is like hoping or moving to the right along the Number Line.

Adding is combining or putting together two or more things (candles, pencils, toys, etc.) or numbers. Numbers to be added are addends and the results of the additions are called sums: 2 + 4 = 6 ; 1 + 3 = 4; 2 + 3 = 5.

Addition II. 2 Digit Addition

Adding double-digit numbers is not as complicated as it looks or sounds. It is easy peasy lemon squeezy. Just follow these instructions to the letter, e.g., 31 + 45 Firstly, line the numbers up correctly:

   31
+ 45
------
And then add the columns from right to left. You have to start by adding the numbers in the “ones” or the units column (1, 5), place the answer at the bottom of the units column, and then move to add the numbers in the “tens” column (3, 4), the tens are written on the second column.

   31
+ 45
------
   76

However, what happens when the sum from the “ones” column is greater than nine? Let’s say 37 + 85, 7 + 5 ( = 12 ) is greater than 10 (12 > 10).

Carry the one over the next column on the left (tens) and write two at the bottom of the units column. When numbers add up to more than ten, the tens need to be carried over to the next column to be added. 1   

   37
+ 85
------
 122

Another way of thinking about it is this: Round one or both addends before adding, thus: A. 40 (37 + 3) + 85 = 125, and then compensate the result. B. 125 - 3 = 122.

TuxMath is a free arcade game that allows you to practice simple arithmetic operations, such as addition, subtraction, multiplication, and division.

Substraction I.

Simplicity is about subtracting the obvious and adding the meaningful, John Maeda.

Do you remember the story of Snow White? The story goes like this: “So the dwarves went hunting without Snow White and a wicked witch came to her. -Have an apple, said the wicked witch. And she handed her an enchanted apple that made her fall asleep.”

You may say: What’s the point in that fairytale? However, it turns out that apples are essential for learning subtraction. How many apples do we have?

That’s right! We have 4 apples. Let’s imagine that a wicked witch has poisoned one of these tasty apples or a nasty worm has ruined an apple, how many edible apples are left?

Exactly, there are 4 - 1 = 3 edible apples left. That’s it! It is as simple as that.

Let’s see another example. Your mum has bought you seven sweets. Your best friend comes to your house and you offer him three. He works like a grasshopper but eats like a ravenous wolf. How many sweets do you have left? 7 - 3 = 4. You have four sweets left.

Imagine that you have eight bones and three dogs steal one bone each from you. How many bones do you have left? 8 -3 = 5 . Of course, you have 5 bones left.

If you want to add numbers together, you jump forwards on the Number Line. On the contrary, if you want to subtract numbers, you jump backwards as you can see in the screenshot below.

Subtraction II. 2 Digit Subtraction.

Subtracting double-digit numbers is not very difficult, but it requires some skill. Just follow these instructions. Let’s subtract 45 - 31. Firstly, line the numbers up correctly

  45
- 31
------

And then, subtract the columns from right to left. You have to start by subtracting the units, the numbers in the “ones” column (5, 1) and then, move on to subtract the numbers in the “tens” column (4, 3).

  45
- 31
------
  14

However, what happens when you can’t subtract, because we all know that 7 is bigger than 5, don’t we?

  85
- 37
------

Just make the “5” larger by borrowing or carrying from the “tens” column. This is called regrouping and you should understand that the number (85 = 70 + 15) remains the same. The calculation becomes 15 - 7 = 8.

It is like having eighty-five dollars. Having eight ten-dollar bills plus five one-dollar bills is the same as having seven ten-dollar bills plus fifteen one-dollar bills.

  85
- 37
------
  48

Now, it’s your turn! Calculate 45 - 32, 24 - 12, 87 - 43, 23 - 19, 93-28, etc.

Multiplication. What is it? How does it work?

Multiplication is basically repeated addition or adding equal groups together. For example, 4 multiplied by 3, 4 x 3 = 4 + 4 + 4 = 12. It equals 12. In other words, the number or factor 4 is added three times.

Observe the following screenshot. We have three groups of five seals. Think about them together as a single group. How many seals are there in total?

That’s right! We have 15 seals. We express this idea as: 5 * 3 = 15.

Another way of thinking about it is as follows. Imagine that your family (your Mum, your Dad, your sister, and you -4-) is having lunch and you have a yummy portion of a pizza in front of you and everyone else, too. Now, suppose for a moment that each one of your family invites two friends.

How can you share the pizza? It is easy, isn’t it? Just split each portion into three. Mathematically, we could say that 4 * 3 = 12 pizza portions. If everyone is still hungry and wants a whole pizza each, you will need to order 12 pizzas because you are twelve. It is another way of looking at it.

A multiple is the result of multiplying a number by an integer. In our example, 12 is a multiple of 3. A times table is a chart or list of multiples of a given number.

Multiplication II. Times tables.

Do you have problems memorizing the multiplication tables? Do not worry my friend, you are in the right place!

0 times table. Anything times zero is zero.

1 times table. Anything times 1 is itself. 1x0=0, 1x1=1, 1x2=2, 1x3=3, 1x4=4, 1x5=5, 1x6=6, 1x7=7, 1x8=8, 1x9=9, 1x10=10.

2 times table. Anything times 2 is doubled and has to end in an even number, 3 * 2 = 3 + 3 = 6. 2x0=0, 2x1=2, 2x2=4, 2x3=6, 2x4=8, 2x5=10, 2x6=12, 2x7=14, 2x8=16, 2x9=18, 2x10=20.

3 and 4 times tables. You need to learn to count by three and four respectively.

3x0=0, 3x1=3, 3x2=6, 3x3=9, 3x4=12, 3x5=15, 3x6=18, 3x7=21, 3x8=24, 3x9=27, 3x10=30.

4x0=0, 4x1=4, 4x2=8, 4x3=12, 4x4=16, 4x5=20, 4x6=24, 4x7=28, 4x8=32, 4x9=36, 4x10=40.

5 times table. Anything times 5 has to end in either a zero or a five. 5x0=0, 5x1=5, 5x2=10, 5x3=15, 5x4=20, 5x5=25, 5x6=30, 5x7=35, 5x8=40, 5x9=45, 5x10=50.

10 times table. Anything times ten is itself and add a zero. 10x0=0, 10x1=10, 10x2=20, 10x3=30, 10x4=40, 10x5=50, 10x6=60, 10x7=70, 10x8=80, 10x9=90, 10x10=100.

Let’s see the most difficult part. Every finger represents a number. Your thumb is the six, the index finger is the seven, and so on. We are going to multiply seven by seven. Then, the index finger (the finger number seven) of one hand has to touch the finger number seven of the other hand. These fingers (both index fingers) with all the fingers beneath them (both thumbs) are tens, thus we have 40 (two index fingers and two thumbs).

Next, you multiply one hand’s remaining fingers by the other hand’s remaining fingers: 3 * 3 = 9, so 7 * 7 = 40 (four index fingers and four thumbs) + 9 = 49.

Try 8 * 7. One hand’s middle finger (8) has to touch the other hand’s index finger (7). These fingers and all the fingers under them are tens, and therefore we have 50 (two thumbs, two index fingers, and one middle finger). Then, we multiply the first hand’s remaining fingers by the second hand’s remaining fingers: 2 * 3 = 6. It is very easy, isn’t it? 8 * 7 = 50 + 6 = 56.

Multiplication III. 2 Digit Multiplication.

Multiplying double-digit numbers is a very straight forward process as I am going to illustrate to you right now. Let’s calculate 31 * 45. Firstly, you need to line the numbers up correctly.
   31
* 45
------
Remember: 45 = 40 + 5, and therefore, 31 * 45 = 31 * (40 + 5) = 31 * 40 + 31 * 5. Then, we calculate 31 * 5. 31  * 45
   31
* 45
------
 155
Next, we calculate 31 * 40. It is just multiplying 31 by 4 and adding a 0 on the final result (31 * 40 = 31 * 4 * 10). 31  * 45
    31
*  45
------
  155
1240   

Finally, we add up both partial results: 31 * 5 and 31 * 40, and we obtain 31 * 45.

       31
*     45
---------
     155
+ 1240  
---------
   1395

That’s right, 31 * 45 = 1395. Now, you should practice these multiplications: 34 * 43, 74 * 49, 14 * 58, and 28 * 76.

Division. What is it? How does it work?

Imagine that we have fifteen yummy bones 15 and three hungry dogs .

How do we share these bones? Of course, we want to be fair, every dog should have the same amount of bones. What is the right answer?

You’ve guessed it! You should give five bones to each dog. Mathematically, we express it as 15 ÷ 3 = 5.

Division is sharing, splitting or grouping a number of things into equal parts or groups.

Besides, there is another way of thinking about this problem. You have three dogs and you want to give them five bones each. How many bones do you need? That’s right! You need 3 * 5 = 15 bones. In other words, division is the inverse or reverse of multiplication.

Division II. Repeated subtraction. Remainder.

Our original division 15 ÷ 5 = 3 can be reinterpreted as repeated subtraction. You can subtract (or you can jump backwards in the Number line) 3 from 15 five times!

Let’s define some concepts now. The dividend is the number that is being divided (15). The divisor is the number dividing it (3). The quotient is the result, which is the number of times that the divisor is contained in the dividend (5).

Sometimes things are not so easy. Imagine a relatively similar problem. There are sixteen bones, just one more 16 and we have three hungry dogs .

How do we split them? The only fair solution is to give each dog five bones and there will still be one bone left. The remainder is what is left over after dividing numbers that do not divide exactly.

It is always the case that: Dividend = Divisor * Quotient + Remainder. In our example, 16 = 3 * 5 + 1.

Division III. 2- or 3-Digit Division

Let’s see how divisions by two or three-digit numbers are done! Let’s calculate 525 ÷ 3.

• First things first, we rewrite 525 ÷ 3 as:

• We look at the first digit of the dividend (5): 5 ÷ 3 = 1 and the remainder equals 2. In other words, how many times will 3 go into 5? The answer is 1, so we put 1 right above the 5. 1*3 = 3, and we do the subtraction 5 - 3 = 2 to get the remainder.

• Let’s move on to the second digit of the dividend (2). Take the elevator and drag it down. We divide the number that we obtained (22), by the divisor (3) and thus: 22 ÷ 3 = 7 and the remainder equals 1: 22 -21 (=3*7) = 1.

• Let’s go to the last digit of the dividend (5). Take the elevator and bring it down again. By doing so, we get 15. We divide this number by the divisor: 15 ÷ 3 = 5 and there is no reminder, so we are done!

Thus, 525 ÷ 3 = 175. We also calculate 276 ÷ 13 = 21 and remainder 3 and 976 ÷ 21 = 46 and remainder 10. Observe that 175 * 3 = 515, 13 * 21 + 3 = 276, and 46 * 21 + 10 = 976.