Remember when you became able to write numbers in your childhood. You wrote “3” if there were three items and wrote “5” if there were five items without any doubt. Maybe you were able to say (count) “ten” for ten items, however, were you able to write it as “10” without any difficulty?

We usually express numbers using only ten characters (so called numeric characters) which are 0-9. In this section, consider carefully how we express (are expressing) numbers of more than ten using only the same ten characters (numeric characters).

In this section, let’s assume that there are ten characters (numeric characters) to express number in the world, which are 0-9 (assume that you live in such a world).

As shown below, we have no difficulty in counting things up to 9.

However, there are only 10 characters (numeric characters), so after 9, we do not have any character to express number.

We can express it if we create a character (numeric character) to express 9 + 1 (such as “ten”), however, we are to express numbers using only ten characters (numeric characters), so it is against the rule.

So let’s make the number that is 1 bigger than 9 (which is the same as the number of characters we can use) as a unit and express it as “10” (read it as “one, zero”).
(Note) Be sure to read it as “one, zero”, not “ten.”

We can take this “10” as there is 1 piece of a unit that expresses the number that is 1 bigger than 9 and 0 piece (nothing) in a part that expresses the number between 0 and 9.
Now if there are 2 pieces of “ten” we can write it as “20,” and if there are 3 pieces of “ten” and 3 pieces of a single item (apple in the figure) we can write it as “32” (in the figure, 3 pieces of bags and 2 pieces of a single item).

Then if the number of “10” (number of bags in the figure) becomes bigger than 9, by introducing “100 (read it “one, zero, zero),” which expresses that there are “10” pieces of “10” (they are shown as a box in the figure), we can count numbers higher than this.

As described above, the method of counting numbers using ten characters (numeric characters), which are 0-9, is called the “decimal notation.”
In addition to these, the number of characters (more precisely, kinds of numbers that can be expressed by 1 digit) is called the “base number.”

In our life, we mostly use decimal numbers and each digit (unit) has its own name. To explain using the previous figure, “bag” is called “ten,” “box” is called a “hundred” (as you know). So, to express “1 (box) 3 (bags) 6 (pieces)” in the world of decimal notation, it is “One hundred and thirty six.”

If there are only four characters (numeric characters) in the world, which are 0-3, how can we count numbers?

We can count number in the same way we do in the world where there are ten numeric characters (decimal notation), which are from 0 to 9 (quaternary notation).
   (However, the boxes and bags are small since the base number is 4)

In the same way, we can count numbers in a world where there are only two characters (numeric characters) which are 0 and 1.

However, the boxes and bags are much smaller since the base number is two (binary notation).

Duodecimal notation and sexagesimal notation for time

To express time, we use a duodecimal notation and a sexagesimal notation. It becomes a little complicated since we use numbers between 0 and 9, which are for the decimal notation we usually use.

One hour is 59 minutes + 1 minute, so as you see, we cannot express minutes using only 1 digit. If we can express from 0 to 59 using only 1 character (1 digit), the calculation of time would be much simpler.

In the world of the PC, sometimes hexadecimal notation is used.
Although the concept is the same in every notation, in hexadecimal notation we express numbers using sixteen kinds of characters that are different from the duodecimal notation and sexadecimal notation.

Normal numeric characters are used for 0 to 9, and for numbers higher than that, a to f are used. “10” in decimal notation is expressed as “a” in hexadecimal notation and “14” in decimal notation is expressed as “e” in hexadecimal notation.

“Exponentiation” is the repeated multiplication of the same numbers, or the number obtained from this.

Ａⁿ is defined as the number obtained by multiplying A n times, which is called A to the n-th power.
For example, 7⁴ represents the number obtained by multiplying 74 times, which means 7 x 7 x 7 x 7 and is called 7 to the 4th power.

Ａ² represents A x A, however Ａ¹(A to the first power) represents A itself.

Now, what does Ａ⁰(A to the 0th power) represent? It represents 1 (one). For any A (precisely any real number) except 0, it is always 1. You can think of it as being defined as so. Once again, Ａ⁰ ＝ 1 , no matter what the value of A is (except A=0).

If a person who lives in the world of N notation can live only using the N notation, there is no problem, however, sometimes it is necessary to convert them.

As we are used to life using the decimal notation, it is difficult to get a sense of numbers expressed in the binary notation or hexadecimal notation which are commonly used in the world of PCs, but sometimes we have to express a number represented in decimal notation in binary notation or hexadecimal notation.

Previously we were considering numbers within each world where only one type of N notation is used, however, if we deal with numbers with different base numbers at the same time, we need to be careful. If we just write “101,” we cannot distinguish whether it is “101” in the decimal notation (so called one hundred and one), or it is “101” in binary notation (“five” in decimal notation).

Given this fact, now we add the base number to the right bottom of the number using brackets. For example, “101” in decimal notation is expressed as “101_(ten)” and “101” in binary notation is expressed as “101_(two)”. If there is nothing written, it is to be taken as being expressed in decimal notation.

Consider how to convert them using the figure shown below, which was used in the explanation of binary notation.

◎ We can express numbers in decimal notation in binary notation by using the method shown below.
     (The direction of the conversion is the same as the direction of the arrow in the figure)

1. First, make a group (divide the number) using the base number (bag), then you get a reminder of it. That is the number for the ones digit (number for a single apple).





2. Next, make a unit of the base number (bag) by the base number (box). The remainder of it is the number for the tens digit (bags left). (There is no bag left in the example above.)





3. In the same way, by making units according to the base number and obtaining the remainder every time, you can obtain the number for the digits in the order from the ones digit.
4. If there is nothing to divide (when the quotient is 0), this means you have completed the conversion and its remainder will be the number for the highest digit.

(Example 1)
As a concrete example, consider how
“to express 434 (four hundred and thirty four) in decimal notation in binary notation.”
To express a number in decimal notation in N notation, as described above, you can do it by dividing the number in decimal notation by N and obtain its remainder from the ones digit to the highest digit, therefore, you can calculate it as shown below. In this case we express it in binary notation, so N=2 (repeat dividing by 2).
      2)434
      2)217　…　0    (the smallest digit in binary notation)
      2)108　…　1    (the second smallest digit in binary notation)
      2) 54　…　0    (the third smallest digit in binary notation)
      2) 27　…　0    (the fourth smallest digit in binary notation)
      2) 13　…　1    (the fifth smallest digit in binary notation)
      2)  6　…　1    (the sixth smallest digit in binary notation)
      2)  3　…　0    (the seventh smallest digit in binary notation)
      2)  1　…　1    (the eighth smallest digit in binary notation)
          0　…　1    (the ninth smallest digit in binary notation) 
Therefore, to express 434 (four hundred and thirty four) expressed in decimal notation in binary notation, it is:
“110110010”

◎ To express a number in binary notation in decimal notation, you can do it as shown below.
     (The direction of the conversion is the reverse of the direction of the arrow in the figure.)

• The weights of each digit in the number expressed in binary notation are as shown below in order from the ones digit.
1. Base number two to the 0th power (one) (number of apples not in a box or anything)
2. Base number two to the first power (two (number of apples in a bag))
3. Base number two to the second power (four (number of apples in a box))
4. Base number two to the third power (eight)
5. ...
• Therefore, it is easy to convert B _(two) (= ...B₃ B₂ B₁ B₀ )in binary notation into D ₍₁₀₎in decimal notation:
  

  D _(ten) ＝ ... +  B₃×2³  +  B₂×2²  +  B₁×2¹  +  B₀×2⁰

  
  
  In the figure above, B _(two)＝101, so you can obtain D _(ten)expressed in decimal notation in the way shown below:

  D _(ten) ＝  1×2²  +  0×2¹  +  1×2⁰ ＝ 4 + 0 + 1 ＝ 5

(Example 2)
As a concrete example, consider how
“to express 110110010 that is expressed in binary notation in decimal notation”
using the result of (Example 1).
To express a number expressed in N notation in decimal notation, do it in the way as described above, which is expressing the weight of each digit in decimal notation, multiplying each digit by the weight, then adding all of them, as in the calculation shown below. In this case, we express a number that is expressed in binary notation in decimal notation, so the weight of the n-th digit from the smallest digit is 2^n-1.
  1×2⁸ +1×2⁷ +0×2⁶ +1×2⁵ +1×2⁴ +0×2³ +0×2² +1×2¹ +0×2⁰
＝ 256 + 128 + 0 + 32 + 16 + 0 + 0 + 2+ 0
＝ 434
Thus, 110110010 expressed in binary notation is expressed in decimal notation as
“434”
Now the number is returned to the number in decimal notation we used in (Example 1).

(Example of the answer)

       
    054-T3298   Senta Sojyo

1. 98 (十)  ==>   1101110 (two) 

(However, the calculation written above is not correct on purpose. Students who have to use 98 should calculate it.)