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St John’s College
The purpose of these activities is to aid you in learning:
-
about Natural numbers, factorising numbers, counting cycles and number bases,
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a little of the history of number, and
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the important mathematical ability of reading a mathematical definition, interpreting the
definition and creating examples from your understanding of the given definition.
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Interesting Natural Numbers
1, 2, …
3. Number with some strange properties
Happy numbers
Happy numbers are produced according to the following rule
Steps:
1. Take each digit of the number and square the digit. (e.g. 15 gives 12 and 52)
2. Add up all the squares of the digits (e.g. 12 +52 = 26)
3. Repeat steps 1 and 2 on the new number (e.g. 26)
4. If the answer becomes a 1 at some point then the number is a Happy
number otherwise the number is an Unhappy number.
a.
b.
c.
d.
Show that 1, 7, 13, 19 and 32 are all Happy numbers.
Show that 2, 5 and 37 are all Unhappy numbers.
What happens to the sequence of numbers produced by steps 1 and 2 for Unhappy numbers?
Give one Happy number greater than 100.
Unhappy numbers
Definition: Not a happy number
Lucky numbers
Write down all the odd numbers 1, 3, 5, 7, 9, … . The first odd number (>1) is 3. Therefore cross
out every third number from the list. Using the new list, the first odd number greater than 3 is 7 so
cross out every seventh number. Continue this process with each new list of numbers. Numbers
remaining after this process is complete are called Lucky numbers.
e. Find the first ten lucky numbers.
Evil numbers
Definition: A Real number in which the first n decimal digits after the comma add up to 666.
Example: √3 (add up the first 146 digits after the comma and you get 666)
f. Look up the digits of π on the Internet and show that π is evil.
Other numbers you can look up:
•
•
•
Rough numbers
Weird numbers
Smooth numbers
http://mathworld.wolfram.com/topics/SpecialNumbers.html
St Johns College, 2006
Interesting Counting Cycles
1, 2, …
3. Mayan Numbers
Two thousand years ago the Maya of Central America used bars and dots to make their written
numbers. They worked in collections of 5 and 20 to make their numbers. Large numbers were
made by placing the bars and dots in a vertical stacks.
0
(an empty shell)
For example, 67 would be written:
1
3 × 20 = 60
+ .7
67
5
20
1 dot × 20
=
20
+ empty shell
=
0
20
Questions
1. Write the following numbers in Mayan symbols
a. 8
b. 16
c. 34
d. 81
e. 100
f. 232
2. Translate the following Mayan numbers into Hindu-Arabic numbers
a.
b.
c.
d.
3. Try to work out our number for the following Mayan number:
St Johns College, 2006
Interesting operations
1. Multiplication in Ancient Egypt
Ancient Egypt was a highly developed civilisation. It used a form of writing known as hieroglyphics
and it also developed a number system. The Egyptian also used arithmetic operations of addition,
subtraction, multiplication and division.
In this worksheet we will explore the method that was used for multiplication. To perform
Egyptian multiplication all that is required is knowledge of addition and the 2 times table.
Example
Find the product of 34 and 27
This column keeps track of the
doubling process.
E.g. 4 is the double of 2
32 is bigger than 27 so we stop.
1
34
2
4
8
16
32
68
136
272
544
This column is the successive
doubling of 34.
E.g. 136 is the double of 68
•
Find a combination of 1, 2, 4, 8, and 16 from the first column that adds up to 27.
•
Now add up the corresponding numbers in the right hand column.
•
•
This gives you the answer to 34 × 27.
Don’t believe me? Check your answer using the multiplication you are used to.
Questions
1. Find the products of the following numbers using Egyptian multiplication.
a. 12 × 15
b. 32 × 75
c. 65 × 87
d. 104 × 321
e. 86 × 54
St Johns College, 2006