Your cart is empty
The decimal system
Ben and Zack collect old coins. They both count how many coins they have. Who chose a smarter system to count them?
When counting coins, it is best to place the coins in groups of ten, then make groups of ten from these groups and so on.
The decimal system is based on grouping by ten.
Positional notation in the decimal system
When counting, you can arrange objects in groups of two, three, four, etc. As a result, there are base \latex{ 2 }, base \latex{ 3 }, base \latex{ 4 }, etc. numeral systems.
The place values you learned in lower grades are units, tens, hundreds, thousands, ten-thousands and hundreds-thousands. To write down large numbers, the place-value table must be expanded.
...hundredstensunitshundredstensunitshundredstensunitshundredstensunits...billion
million
thousand
Take a look at the number \latex{ 2,209 }. Why does the digit \latex{ 2 } have various real values?
Digitsplace valueface valuereal value\latex{2}
\latex{2}
\latex{0}
\latex{9}
thousands
\latex{2}
\latex{2,000}
hundreds
\latex{2}
\latex{200}
tens
\latex{0}
\latex{0}
units
\latex{9}
\latex{9}
\latex{2}\latex{,}\latex{2}\latex{0}\latex{9}
face value:
real value:
place value:
hundreds
\latex{2}
\latex{200}
Decomposing \latex{ 2,209 } by place value:
\latex{2,209 = 2 \text{ thousands} + 2 \text{ hundreds} + 0 \text{ tens} + 9 \text{ units}}
\latex{2,209 = 2 \times 1,000 + 2 \times 100 + 0 \times 10 + 9 \times 1}
Large numbers can be decomoposed in a similar way:
\latex{\textbf{3,752,194} = \textbf{3} \text{ millions} + \textbf{7} \text{ hundred-thousands} + \textbf{5} \text{ ten-thousands} + \textbf{2} \text{ thousands}}Â \latex{+ \textbf{1} \text{ hundreds} + \textbf{9} \text{ tens} + \textbf{4} \text{ units}}
\latex{\textbf{3,752,194} = \textbf{3} \times 1,000,000 + \textbf{7} \times 100,000 + \textbf{5} \times 10,000 + \textbf{2} \times 1,000 + \textbf{1} \times 100 + \textbf{9} \times 10 + \textbf{4} \times 1}
Multiple-digit numbers can be organised in groups consisting of \latex{ 3 } digits to make interpretation easier (see place-value table):
\latex{ 3 }\latex{ , }\latex{ 752 }\latex{ , }\latex{ 194 } = \latex{ 3 } million \latex{ 752 } thousand \latex{ 194 }
Natural numbers composed of two, three, four, etc. digits are called two-digit, three-digit, four-digit, etc. numbers.
There are \latex{ 10 } single-digit natural numbers: {\latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 }; \latex{ 5 }; \latex{ 6 }; \latex{ 7 }; \latex{ 8 }; \latex{ 9 }}.
Example
How many positive two-digit whole numbers are there?
Solution 1
tens
units
nine different digits
ten different digits
Write down the positive whole numbers from \latex{ 1 } to \latex{ 99 }.
\latex{{\underbrace{1; 2; 3; 4; 5; 6; 7; 8; 9}_{\text{single-digit: } 9} \enspace \underbrace{10; 11; ...; 42; 43; ...; 97; 98; 99}_{\text{ two-digit: } 99 - 9 = 90} }}
There are \latex{ 99 } positive whole numbers that consist of one or two digits. Nine of them consist of one single digit, meaning that there are \latex{ 90 } positive two-digit whole numbers.
tens units number
\latex{19}\latex{ 1 }\latex{0}\latex{10}\latex{11}\latex{12}\latex{18}\latex{1}\latex{9}\latex{8}\latex{2}Solution 2
There are \latex{ 9 } digits that can go in the tens place (since a two-digit number cannot start with zero). There are \latex{ 10 } digits that can go in the place of the units (even \latex{ 0 }).
Therefore, there are \latex{9 \times 10 = 90} two-digit positive whole numbers.
..\latex{99}\latex{ 9 }\latex{0}\latex{90}\latex{91}\latex{92}\latex{98}\latex{1}\latex{9}\latex{8}\latex{2}Writing natural numbers
Numbers can be written with digits or with letters. There are rules for both.
WITH DIGITS:
In the case of multi-digit numbers, a comma is added to every third number starting from the units. For example \latex{ 1,848 }; \latex{ 12,002 }; \latex{ 5,273,964 }.
In the case of multi-digit numbers, a comma is added to every third number starting from the units. For example \latex{ 1,848 }; \latex{ 12,002 }; \latex{ 5,273,964 }.
IN WORDS:
When writing numbers with letters, a hyphen should be added to numbers between \latex{ 21 } and \latex{ 99 } (e.g. \latex{ 67 }: sixty-seven). In the case of four-digit numbers or larger, simply add a comma after every third number starting from the right.
For example: Twelve thousand two (\latex{ 12,002 }); two hundred sixty-three thousand five hundred sixty-four (\latex{ 263,564 }).
Exercises
{{exercise_number}}. How many €\latex{ 10 } banknotes are the following amounts?
a) €\latex{ 700 }
b) €\latex{ 4,000 }
c) €\latex{ 50,000 }
{{exercise_number}}. How many two-digit numbers are there where the tens digit is larger than the units digit by two?
{{exercise_number}}. How many three-digit numbers are there where the largest digit is in the smallest place value position? Write them down. Write down the smallest number with letters as well.
{{exercise_number}}. How many tens are in
a) \latex{ 3 } hundreds;
b) \latex{ 7 } thousands;
c) \latex{ 1 } ten-thousand?
{{exercise_number}}. How many €\latex{ 100 } banknotes would you get for the following amounts?
a) €\latex{ 2,500 }
b) €\latex{ 13,500 }
c) €\latex{ 30,600 }
d) €\latex{ 84,700 }
{{exercise_number}}. How many tens are in
a) \latex{ 3 } thousands;
b) \latex{ 9 } thousands;
c) \latex{ 2 } ten thousands?
{{exercise_number}}. How many tens are in
a) \latex{ 4 } ten thousands and \latex{ 7 } hundreds;
b) \latex{ 8 } ten thousands and \latex{ 15 } thousands;
c) \latex{ 1 } hundred thousands and \latex{ 64 } thousands?
{{exercise_number}}. You have four number cards:
\latex{2}\latex{5}\latex{7}\latex{9}a)Â How many four-digit numbers can you make using them?
b) How many of these are even? (→)
HELP • for Exercise \latex{8}
Three-digit numbers you can make using the digits \latex{1}; \latex{2} and \latex{3}:
{{exercise_number}}. You can use the digits \latex{ 2 }, \latex{ 5 }, and \latex{ 7 } as many times as you want to make three-digit numbers.
a) How many different three-digit numbers can we make?
b) How many of these are odd?
b) How many of these are odd?
{{exercise_number}}. How many tens are in
a) \latex{ 18,400 }
b)Â \latex{ 50,116 }
c) \latex{ 6,001 }
d) \latex{ 348 }
e) \latex{ 1,256 }
f) \latex{ 4,372 }
g) \latex{ 2,000 }
h)Â \latex{ 32,100 }
{{exercise_number}}. Write down an eight-digit number in which the digit found in the tens place is one greater than the digit found in the units place, the digit found in the hundreds place is one greater than the digit found in the tens place, and so on. How many such numbers are there? Write down the smallest with letters as well.
{{exercise_number}}. When did the following animals appear on Earth? Write down the years using digits.
a)Â First life forms, more than \latex{ 3 } billion years ago
b) First fish, about \latex{ 600 } million years ago
c)Â First dinosaurs, about \latex{ 400 } million years ago
d)Â Birds, about \latex{ 200 } million years ago
e)Â Present-day man, about \latex{ 100 } thousand years ago
b) First fish, about \latex{ 600 } million years ago
c)Â First dinosaurs, about \latex{ 400 } million years ago
d)Â Birds, about \latex{ 200 } million years ago
e)Â Present-day man, about \latex{ 100 } thousand years ago
{{exercise_number}}. How many natural numbers have
a) three digits;
b) four digits;
c) five digits?
Quiz
How many three-digit palindromic numbers are there? (Palindromic numbers are numbers that remain unchanged when their digits are reversed. Example: \latex{ 202 })