Cosc 2150: Computer Organization Chapter 2 Part 1 Integers addition and subtraction.
Modeling Addition of Integers
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Transcript of Modeling Addition of Integers
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Modeling Addition of Integers
+ – + –
+ –
To model addition problems involving positive and negative integers, you can
use tiles labeled and . Each represents positive 1, and each
represents negative 1. Combining a with a gives 0. You can use
algebra tiles to find the sum of –8 and 3.
1 Model negative 8 and positive 3 using algebra tiles.
+– – – – – – – – + +
2 Group pairs of positive and negative tiles. Count the remaining tiles. +
– – – – – – – –
+ +
3 The remaining tiles show the sum of –8 and 3.
–8 + 3 = –5
3
Each pair has a sum of 0.Each pair has a sum of 0.
– – – – –
–80
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The sum can be written as –2 + 5 = 3.
Addition can be modeled with movements on a number line.
You add a positive number by moving to the right.Model –2 + 5.
1–3 –2 –1 0 32– 4 4
Move 5 units to the right.Start at –2. End at 3.
• •
Adding Real Numbers
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The sum can be written as –2 + 5 = 3.
Addition can be modeled with movements on a number line.
You add a positive number by moving to the right.
You add a negative number by moving to the left.
Model –2 + 5.
1–3 –2 –1 0 32– 4 4
0– 4 –3 –2 –1 21–5 3
Move 5 units to the right.
Move 6 units to the left.
Model 2 + (–6).
Start at –2. End at 3.
• •
Start at 2.End at – 4.
••
Adding Real Numbers
The sum can be written as 2 + (–6) = – 4.
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The rules of addition show how to add two real numbers without a number line.
Adding Real Numbers
RULES OF ADDITION
TO ADD TWO NUMBERS WITH THE SAME SIGN:
1 Add their absolute values.
2 Attach the common sign.
Example: – 4 + (–5) – 4 –5+ = 9 –9
![Page 5: Modeling Addition of Integers](https://reader036.fdocuments.in/reader036/viewer/2022081821/56812cb1550346895d9161b3/html5/thumbnails/5.jpg)
The rules of addition show how to add two real numbers without a number line.
Adding Real Numbers
RULES OF ADDITION
TO ADD TWO NUMBERS WITH THE SAME SIGN:
1 Add their absolute values.
2 Attach the common sign.
Example: – 4 + (–5) – 4 –5+ = 9 –9
TO ADD TWO NUMBERS WITH OPPOSITE SIGNS:
1 Subtract the smaller absolute value from the larger absolute value.
2 Attach the sign of the number with the larger absolute value.
Example: 3 + (–9) –9 3– = 6 –6
![Page 6: Modeling Addition of Integers](https://reader036.fdocuments.in/reader036/viewer/2022081821/56812cb1550346895d9161b3/html5/thumbnails/6.jpg)
The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition.
Adding Real Numbers
PROPERTIES OF ADDITION
COMMUTATIVE PROPERTY
The order in which two numbers are added does not change the sum.
a + b = b + a Example: 3 + (–2) = –2 + 3
ASSOCIATIVE PROPERTY
The way you group three numbers when adding does not change the sum.
(a + b) + c = a + (b + c) Example: (–5 + 6) + 2 = –5 + (6 + 2)
![Page 7: Modeling Addition of Integers](https://reader036.fdocuments.in/reader036/viewer/2022081821/56812cb1550346895d9161b3/html5/thumbnails/7.jpg)
The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition.
Adding Real Numbers
PROPERTIES OF ADDITION
IDENTITY PROPERTY
The sum of a number and 0 is the number.
a + 0 = a Example: – 4 + 0 = – 4
PROPERTY OF ZERO (INVERSE PROPERTY)
The sum of a number and its opposite is 0.
a + (–a) = 0 Example: 5 + (–5) = 0
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Adding Three Real Numbers
Use a number line to find the following sum.
–3 + 5 + (– 6)
SOLUTION
0– 4 –3 –2 –1 21–5 3
Move 6 units to the left.
–3 + 5 = 2
End at – 4.
••Start at –3. Move 5 units to the right.
The sum can be written as –3 + 5 + (–6) = – 4.
•
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Finding a Sum
Find the following sums.
1.4 + (–2.6) + 3.1 Use associative property.= 1.4 + (–2.6 + 3.1)
Simplify.= 1.4 + 0.5
= 1.9
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Finding a Sum
Find the following sums.
1.4 + (–2.6) + 3.1 Use associative property.= 1.4 + (–2.6 + 3.1)
Simplify.= 1.4 + 0.5
= 1.9
– + 3 + 12
12
= – + + 312
12
– + + 312
12( ) =
= 0 + 3 = 3
Use commutative property.
Use associative property.
Use identity property and property of zero.
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Using Addition in Real Life
SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions?
Aluminum: 13 electrons, 13 neutrons, 13 protons
SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al.
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Using Addition in Real Life
SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions?
Aluminum: 13 electrons, 13 neutrons, 13 protons
Aluminum: 10 electrons, 13 neutrons, 13 protons
SOLUTION The total charge is –10 + 0 + 13 = 3, so the atom is an ion. In chemistry, this aluminum atom is written as Al
3 +.
SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al.
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Finding the Total Profit
A consulting company had the following monthly results after comparing income and expenses. Add the monthly profits and losses to find the overall profit or loss during the six-month period.
$12,932.54$7613.17$3825.01
JuneMayApril
–$4734.86–$6783.16–$13,142.50
MarchFebruaryJanuary
SOLUTION With this many large numbers, you may want to use a calculator.
13142.50 6783.16 4734.86
3825.01 7613.17 12932.54
+/– +
=
+/– + +/–
+ + + –289.8
The display is –298.8. This means the company had a loss of $298.80.