1.1: Addition of Integers
On Monday, Marty borrows $50.00 from his father. On Thursday, he gives his father $28.00. Can you write an addition statement to describe Marty’s financial transactions?
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Khan Academy Adding/Subtracting Negative Numbers
Guidance
When adding integers, you need to make sure you follow two rules:
- Integers with unlike signs must be subtracted. The answer will have the same sign as that of the higher digit.
- Integers with the same sign must be added. The answer will have the same sign as that of the digits being added.
In order to understand why these rules work, you can represent the addition of integers with manipulatives such as color counters and algebra tiles. A number line can also be used to show the addition of integers. The following examples show how to use these manipulatives to understand the rules for adding integers.
Example A
\begin{align*}5+(-3)=?\end{align*}
The problem can be represented by using color counters. In this case, the red counters represent positive numbers and the yellow ones represent the negative numbers.
The above representation shows the addition of 5 positive counters and 3 negative counters.
One positive counter and one negative counter equals zero. \begin{align*}1+(-1)=0\end{align*}
Draw a line through the counters that equal zero.
The remaining counters represent the answer. Therefore, \begin{align*}5+(-3)=2\end{align*}. The answer is the difference between 5 and 3. The answer takes on the sign of the larger digit and in this case the five has a positive value and it is greater than 3.
Example B
\begin{align*}4+(-7)=?\end{align*}
The above representation shows the addition of 4 positive counters and 7 negative counters.
One positive counter and one negative counter equals zero. \begin{align*}1+(-1)=0\end{align*}
Draw a line through the counters that equal zero.
The remaining counters represent the answer. Therefore, \begin{align*}4+(-7)=-3\end{align*}. The answer is the difference between 7 and 4. The answer takes on the sign of the larger digit and in this case the seven has a negative value and it is greater than 4.
Example C
This same method can be extended to adding variables. Algebra tiles can be used to represent positive and negative values.
\begin{align*}6x+(-8x)=?\end{align*}
The green algebra tiles represent positive \begin{align*}x\end{align*} and the white tiles represent negative \begin{align*}x\end{align*}. There are 6 positive \begin{align*}x\end{align*} tiles and 8 negative \begin{align*}x\end{align*} tiles.
The remaining algebra tiles represent the answer. There are two negative \begin{align*}x\end{align*} tiles remaining. Therefore, \begin{align*}(6x)+(-8x)=-2x\end{align*}. The answer is the difference between \begin{align*}8x\end{align*} and \begin{align*}6x\end{align*}. The answer takes on the sign of the larger digit and in this case the eight has a negative value and it is greater than 6.
Example D
\begin{align*}(-3)+(-5)=?\end{align*}
The solution to this problem can be determined by using a number line.
Indicate the starting point of -3 by using a dot. From this point, add a -5 by moving five places to the left. You will stop at -8.
The point where you stopped is the answer to the problem. Therefore, \begin{align*}(-3)+(-5)=-8\end{align*}
Concept Problem Revisited
On Monday, Marty borrows $50.00 from his father. On Thursday, he gives his father $28.00.
Marty borrowed $50.00 which he must repay to his father. Therefore Marty has \begin{align*}-\$50.00\end{align*}.
He returns $28.00 to his father. Now Marty has \begin{align*}-\$50.00+(\$28.00)=-\$22.00\end{align*}. He still owes his father $22.00.
Vocabulary
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list ..., -3, -2, -1, 0, 1, 2, 3...
- Irrational Numbers
- The irrational numbers are those that cannot be expressed as the ratio of two numbers. The irrational numbers include decimal numbers that are non-terminating decimals as well as those with digits that do not repeat with a pattern.
- Natural Numbers
- The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are numbers in the list 1, 2, 3... and are often referred to as positive integers.
- Number Line
- A number line is a line that matches a set of points and a set of numbers one to one.
It is often used in mathematics to show mathematical computations.
- Rational Numbers
- The rational numbers are numbers that can be written as the ratio of two numbers \begin{align*}\frac{a}{b}\end{align*} and \begin{align*}b \neq 0\end{align*}. The rational numbers include all terminating decimals as well as those decimals that are non-terminating but have a repeating pattern of digits.
- Real Numbers
- The rational numbers and the irrational numbers make up the real numbers.
Guided Practice
1. Use a model to answer the problem \begin{align*}(-7)+(+5)=?\end{align*}
2. Use the number line to determine the answer to the problem \begin{align*}8+(-2)=?\end{align*}
3. Determine the answer to \begin{align*}(-6)+(-3)=?\end{align*} and \begin{align*}(2)+(-5)=?\end{align*} by using the rules for adding integers.
Answers:
1. \begin{align*}(-7)+(+5)=?\end{align*}
Cancel the counters that equal zero
There are 2 negative counters left. Therefore, \begin{align*}(-7)+(+5)=-2\end{align*}. The answer is the difference between 7 and 5. The answer takes on the sign of the larger digit and in this case the seven has a negative value and it is greater than 5.
2. \begin{align*}8+(-2)=?\end{align*}
You begin on 8 and move two places to the left. You stop at 6. Therefore \begin{align*}8+(-2)=6\end{align*}.
The answer is the difference between 8 and 2. The answer takes on the sign of the larger digit and in this case the eight has a positive value and it is greater than 2.
3. \begin{align*}(-6)+(-3)=?\end{align*}
Both numbers have negative signs. The numbers must be added and the sum will be a negative answer. Therefore \begin{align*}(-6)+(-3)=-9\end{align*}.
\begin{align*}(2)+(-5)=?\end{align*}
The numbers being added have different signs. The numbers must be subtracted and the answer will have the sign of the larger digit. Therefore \begin{align*}(2)+(-5)=-3\end{align*}.
Practice
Complete the following addition problems using any method.
- \begin{align*}(-7)+(-2)\end{align*}
- \begin{align*}(6)+(-8)\end{align*}
- \begin{align*}(5)+(4)\end{align*}
- \begin{align*}(-7)+(9)\end{align*}
- \begin{align*}(-1)+(5)\end{align*}
- \begin{align*}(8)+(-12)\end{align*}
- \begin{align*}(-2)+(-5)\end{align*}
- \begin{align*}(3)+(4)\end{align*}
- \begin{align*}(-6)+(10)\end{align*}
- \begin{align*}(-1)+(-7)\end{align*}
- \begin{align*}(-13)+(9)\end{align*}
- \begin{align*}(-3)+(-8)+(12)\end{align*}
- \begin{align*}(14)+(-6)+(5)\end{align*}
- \begin{align*}(15)+(-8)+(-9)\end{align*}
- \begin{align*}(7)+(6)+(-9)+(-8)\end{align*}
For each of the following models, write an addition problem and answer the problem.
Irrational Number
An irrational number is a number that can not be expressed exactly as the quotient of two integers.Natural Numbers
The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are the numbers in the list 1, 2, 3... and are often referred to as positive integers.number line
A number line is a line on which numbers are marked at intervals. Number lines are often used in mathematics to show mathematical computations.operation
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.rational number
A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.Image Attributions
Here you'll learn to add integers using different representations including a number line. These methods will lead to the formation of two rules for adding integers.
Concept Nodes:
Irrational Number
An irrational number is a number that can not be expressed exactly as the quotient of two integers.Natural Numbers
The natural numbers are the counting numbers and consist of all positive, whole numbers. The natural numbers are the numbers in the list 1, 2, 3... and are often referred to as positive integers.number line
A number line is a line on which numbers are marked at intervals. Number lines are often used in mathematics to show mathematical computations.operation
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.rational number
A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.Real Number
A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.