Most notable advancements in the early
development of mathematics:
• Mayans
• Babylonians
• Egyptians
• Greeks
• Chinese
Wrote on tablets
• Used two symbols for numbers
• Ones
• Tens
• Used a base 60 place system
•
•
clocks (60 seconds, 60 minutes or
3600 seconds)
circle (360°)
Tablet with
numbers
1 set of 3600
52 sets of 60
30 sets of 1
1
52
30
1 ˟ 3600 =
52 ˟ 60 =
30 ˟ 1
=
3600
3120
30
6750
Try to write:
23
41
82
121
82 = 60 + 22
121 = 2 ˟ 60 + 1
Babylonian multiplication concentrated on
perfect squares
(3)(4) = (3 + 4)2 – 32 – 42 = 49 – 9 – 16 = 24 = 12
2
2
2
Simple grouping system (hieroglyphics)
The Egyptians used the
stick for 1
heel bone for 10
scroll for 100
lotus flower for 1,000
bent finger for 10,000
burbot fish for 100,000
astonished man
for 1,000,000.
3000 + 200 + 40 + 4
= 3244
What are the
following values?
52
21,238
The Ancient Egyptians
used a pencil and
paper method for
multiplication which
was based on doubling
and addition.
Write down 1 and 50
Work down, doubling the numbers, so
that you’ve now got 2, 4, 8, 16, etc.
lots of 53.
Stop when the number of the left
(16) is more than half of the other
number you are multiplying (18).
Look for numbers on the left that add
up to 18 (2 and 16).
Cross out the other rows of numbers.
Add up the remaining numbers on the
right to get the final answer.
1
2
4
8
16
50
100
200
400
800
900
Write 1 and 76, meaning 1 lot of 76.
Work down, doubling the numbers, so
that you’ve now got 2, 4, 8, 16, etc.
lots of 76.
Stop when the number of the left
(32) is more than half of the other
number you are multiplying (39).
Look for numbers on the left that add
up to 39 (1, 2, 4 and 32).
1
2
4
8
16
32
76
152
304
608
1216
2432
Cross out the other rows of numbers.
Add up the remaining numbers on the
right to get the final answer.
2964
This jar holds 17 litres of water.
How much water will 25 jars hold?
A potter makes 35 pots each month.
How many will he make in a year?
This chariot travels 23km in an
hour. How far will it travel in 6
hours?
We have seen that different civilizations
had different methods to handle basic
arithmetic
This demonstrates that we can
add, subtract, multiply, and
divide numbers in multiple
ways and still get the same
answer
43
+ 25
Add the tens (40 + 20)
Add the ones (3 + 5)
Add the partial sums
(60 + 8)
60
+ 8
68
Add the
Add the
Add the
Add the
hundreds (200 + 400)
tens (60 +80)
ones (8 + 3)
partial sums
(600 + 140 + 11)
268
+ 483
600
140
+ 11
751
1. Create a grid
2. Draw diagonals
3. Add each column, placing
the tens digit in the upper
half of the cell and the ones
digit in the bottom half of
the cell
4. Add along each diagonal
and record any regroupings
in the next diagonal
7
+ 4
1
1
2
1
8
8
1
6
6
The opposite change rule says that if a
value is added to one of the numbers, then
subtract the value from the other number
88
+ 36
+2
-2
90
+ 34
+10
- 10
100
+ 24
124
Let’s look at some different
methods to subtract numbers
We are familiar with the basic
borrowing methods, but did
you know we can subtract by
adding?
1. Place the smaller
number at the bottom of
the hill and the larger at
the top.
38 – 14 =
2. Start with 14, add
to the next friendly
number. (14+6=20)
3. Start with 20,
add to the next
friendly number.
(20+10=30)
4. Start with 30, add
to get 38. (30+8=38)
Record the
numbers added at
each interval:
(6+10+8=24)
1. Replace each digit to
be subtracted with its
nines complement,
and then add
2. Delete the leading 1
3. Add 1 to the final
result
75
– 38
75
+ 61
136
+1
37
Let’s look at some different
methods to multiply numbers
We have already seen two
methods to multiply beyond
our current procedure
(Babylonian method of
squares and the Egyptian
method of doubles. Let’s look
at a few more.
When multiplying by “Partial
Products,” you must first
multiply parts of these numbers,
then you add all of the results to
find the answer.
Multiply 20 X 60 (tens by tens)
Multiply 60 X 7 (tens by ones)
Multiply 4 X 20 (ones by tens)
Multiply 7 X 4 (ones by ones)
Add the results
27
X 64
1,200
420
80
+ 28
1,728
(20+7)
(60+4)
1. Create a grid
2. Draw diagonals
3. Copy one digit across top of
grid and the other along
the right side
4.
5.
25 x 47 = 1,175
2 5
0 2 4
Multiply each digit in the
1
8
0
1
top factor by each digit in
the side factor, placing the
1 3 7
tens digit in the upper half
4
5
1
of the cell and the ones digit
in the bottom half of the cell
Add along each diagonal
and record any regroupings
in the next diagonal
7
5
We can often perform basic
arithmetic in our head faster than
we can by writing it down or
plugging it into a calculator.
We need to recognize certain
patterns to help the process.
We can add
large set of
numbers
quickly by
grouping
values that
add to ten
10
20
10
2
52
47
63
28
+ 16
20 6
10
10
26
6
We can multiply by four simply by
doubling the value twice:
37 x 4
double
115 x 4
double
74
double again
148
230
double again
460
We can multiply by five simply by
multiplying by ten and then take
half:
42 x 5
multiply by 10
multiply by 10
420
take half
73 x 5
730
take half
210
365
We can multiply by eleven by keeping
the first and last digit and then adding
digits that are next to each other to get
the rest of the digits
3+5
35 x 11 = 3 8 5
1+4
4+2
142 x 11 = 1 5 6 2
Keep in mind that there is more
than one way to get to the correct
answer. We have shown you a few
different methods to add, subtract
and multiply, but there are many
other methods.
Try these or other methods to see if
you like them. Perhaps you can
invent your own.