1.3: Multiplication of Real Numbers
Multiplication of Integers
Objectives
The lesson objectives for The Multiplication of Real Numbers are:
- Multiplication of Integers Using Rules
- Multiplication of Fractions Using Rules
- Addition of Decimal Numbers Using the Rules
Introduction
In this concept you will learn to multiply integers, fractions and decimal numbers by using rules. All of these will be presented in one lesson. To become familiar with the processes, you will watch videos that demonstrate each of the objectives.
Watch This
Khan Academy Multiplying and Dividing Negative Numbers
Khan Academy Multiplying Fractions
Khan Academy Multiplication 8: Multiplying Decimals
Guidance
Multiplication by a positive integer can be thought of as repeated addition. To represent this by using color counters, one red counter will be +1 and one yellow counter will be -1.
Jacob received tips of $4.00 each from three of his paper route customers. How much did he receive in total?
To solve this problem, start with 0. (The empty blue rectangle represents 0)
Add three groups of four red counters.
The result of \begin{align*}(+3) \times (+4)\end{align*} is +12. The product of two positive integers is always positive.
Example A
Sam spent $2.00 for a bottle of chocolate milk at the school cafeteria every school day. At the end of the week, how does this affect his net worth?
Start with 0. Add 5 groups of yellow counters.
The result of \begin{align*}(+5) \times (-2)\end{align*} is -10. The product of a positive integer and a negative integer is always negative.
Example B
Multiplication by a negative integer can be thought of as repeated subtraction.
What is the result of \begin{align*}(-2) \times (-3)\end{align*}? Start with equal numbers of red and yellow counters to represent zero. Remove two groups of (-3). The result is +6.
The result of \begin{align*}(-2) \times (-3)\end{align*} is +6. The product of two negative integers is always positive.
You now have the rules for multiplying integers. When you multiply two integers that have the same sign, the product will always be positive. When you multiply two integers that have opposite signs, the product will always be negative.
Example C
i) \begin{align*}\left(\frac{2}{3}\right) \times \left(\frac{5}{7}\right)\end{align*}
ii) \begin{align*}\left(\frac{7}{8}\right) \times \left(3 \frac{3}{4}\right)\end{align*}
iii) \begin{align*}\left(5 \frac{3}{4} \right) \times \left(2 \frac{3}{5}\right)\end{align*}
There are three simple steps to follow to multiply fractions:
- Multiply the numerators of the fractions
- Multiply the denominators of the fractions.
- Simplify the fraction if necessary.
i) \begin{align*}\left(\frac{2}{3}\right) \times \left(\frac{5}{7}\right)\end{align*}
\begin{align*}& \frac{2 \times 5}{3 \times 7}\\ & \frac{10}{21}\end{align*}
ii) \begin{align*}\left(\frac{7}{8}\right) \times \left(3 \frac{3}{4}\right)\end{align*} Express the mixed number as an improper fraction.
\begin{align*}& \left(\frac{7}{8}\right) \times \left(\frac{15}{4}\right) \rightarrow \frac{(4 \times 3)+3}{4}\\ & \frac{7 \times 15}{8 \times 4}\\ & \frac{105}{32}=3 \frac{9}{32} \rightarrow \ \overset{ \quad \ 3}{32 \overline{ ) {105 }}} \quad 105-96=9\end{align*}
iii) \begin{align*}\left(5 \frac{3}{4}\right) \times \left(2 \frac{3}{5}\right)\end{align*} Express the mixed numbers as improper fractions.
\begin{align*}& \left(\frac{23}{4}\right) \times \left(\frac{13}{5}\right) \rightarrow \frac{(4 \times 5)+3}{4} \ \text{and} \ \frac{(5 \times 2)+3}{5}\\ & \frac{23 \times 13}{4 \times 5}\\ & \frac{299}{20}=14 \frac{19}{20} \rightarrow \overset{ \quad 14}{20 \overline{ ) {299 }}} \quad 299-280=19\end{align*}
Example D
\begin{align*}(14.65) \times (2.7)\end{align*}
Multiply the numbers as you would whole numbers. To place the decimal in the answer, count the number of digits after the decimal points in the problem. There are two digits after the decimal point in 14.65 and one digit after the decimal point in 2.7. This is a total of three digits after the decimal points. From the right of the answer, count three places to the left and insert the decimal point.
\begin{align*}& 14.65\\ & \underline{\times \; 2.7 \;\;}\\ & \ \ 10255\\ & \underline{+29300}\\ & \ \ \underset{\quad \ {\color{red}\longleftarrow}}{39 {\color{red}.} 555}\end{align*}
Vocabulary
- Denominator
- The denominator of a fraction is the number on the bottom that indicates the total number of equal parts in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has denominator 8.
- Fraction
- A fraction is any rational number that is not an integer.
- Improper Fraction
- An improper fraction is a fraction in which the numerator is larger than the denominator. \begin{align*}\frac{8}{3}\end{align*} is an improper fraction.
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list ..., -3, -2, -1, 0, 1, 2, 3...
- Mixed Number
- A mixed number is a number made up of a whole number and a fraction such as \begin{align*}4 \frac{3}{5}\end{align*}.
- Numerator
- The numerator of a fraction is the number on top that is the number of equal parts being considered in the whole or the group. \begin{align*}\frac{5}{8}\end{align*} has numerator 5.
Guided Practice
1. Use a model to represent the problem \begin{align*}(-3) \times (+2)\end{align*}
Write a word problem to that could be represented by the model.
2. Multiply the following fractions:
i) \begin{align*}\left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)\end{align*}
ii) \begin{align*}\left(3\frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)\end{align*}
3. Determine the answer to \begin{align*}(-135.697) \times (-34.32)\end{align*}
Answers
1. \begin{align*}(-3) \times (+2)\end{align*}
The model must start with an equal number of red and yellow counters to represent zero.
Three groups of 2 red counters must be removed.
Sherri lost three pairs of earrings through a hole in her jewelry pouch. How does this affect her net worth?
2. \begin{align*}\left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)\end{align*} Multiply the numerators. Multiply the denominators. Simplify the fraction.
Note: Remember the rule for multiplying integers. When you multiply two integers that have opposite signs, the product will always be negative.
\begin{align*}& \left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)\\ & \left(\frac{5}{9}\right) \times \left(\frac{-4}{7}\right)=\frac{5 \times (-4)}{9 \times 7}=-\frac{20}{63}\end{align*}
The answer can be written as \begin{align*}\frac{-20}{63}\end{align*} or \begin{align*}-\frac{20}{63}\end{align*}. The denominator of a fraction cannot be a negative value. If there is a negative sign in the denominator, move it up to the numerator of the fraction. If there are two signs in the numerator, change them to a single sign. You should now know that two opposite signs become a negative sign and two like signs become a positive sign.
\begin{align*}+-=- \qquad --=+\\ -+=- \qquad ++=+\end{align*}
ii) \begin{align*}\left(3 \frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)\end{align*}
Write the two mixed numbers as improper fractions. Multiply the denominator and the whole number. Add the numerator to this product. Write the answer over the denominator. Follow the steps for multiplying fractions. Simplify the fraction if necessary.
\begin{align*}& \left(3 \frac{2}{3}\right) \times \left(4 \frac{1}{5}\right)\\ & \left(\frac{11}{3}\right) \times \left(\frac{21}{5}\right)\\ & \left(\frac{11}{3}\right) \times \left(\frac{21}{5}\right)=\frac{231}{15}\\ & \frac{231}{15}=\overset{ \quad 15}{15 \overline{ ) {231 }}} \quad 15 \times 15=225 \quad 231-225=6\\ & \frac{231}{15}=15 \frac{6}{15}=15 \frac{2}{5}\end{align*}
3. \begin{align*}(-135.697) \times (-34.32)\end{align*}
Multiply the numbers as you would whole numbers. Remember the rule for multiplying integers. When you multiply two integers that have the same sign, the product will always be positive.
\begin{align*}& \ \ -135.697\\ & \underline{\times \; -34.32 \;\;\;\;}\\ & \quad \quad \ \ 271394\\ & \quad \quad \ 407091 {\color{blue}0}\\ & \quad \ \ 542788 {\color{blue}00}\\ & \underline{\;\;\;\; 407091 {\color{blue}000} \;}\\ & \quad \underset{\quad \ {\color{red}\longleftarrow}}{4567{\color{red}.}12104}\end{align*}
There are three digits after the decimal point in 135.697 and two digits after the decimal point in 34.32. Thus is a total of five places. Beginning at the right of the product, count five places to the left and insert the decimal point.
Summary
The multiplication of integers can be represented with manipulative such as color counters. Multiplication by a positive integer can be thought of as repeated addition and multiplication by a negative integer can be thought of as repeated subtraction. The models demonstrated that multiplication of two integers with the same signs produces a positive result and multiplication of two integers with unlike signs results in a negative answer.
These rules can be applied to the multiplication of all real numbers. In this lesson you also learned to multiply proper fractions as well as mixed numbers. To multiply fractions, you multiplied the numerators and then you multiplied the denominators. The product of the numerators over the product of the denominators was the answer to the problem. If the answer could be expressed as an equivalent fraction, you showed this as your answer to the question.
The final topic that was presented in this lesson was the multiplication of decimals. You learned that multiplying the decimal numbers was the same as multiplying whole numbers. The rules for multiplying integers were also applied to multiplying decimal numbers. The sum of the number of digits after the decimal points determined the placement of the decimal point in the answer.
Problem Set
Use models of color counters to represent the following multiplication problems and use that model to determine the answer.
- \begin{align*}(-7) \times (-2)\end{align*}
- \begin{align*}(+3) \times (+4)\end{align*}
- \begin{align*}(-5) \times (+3)\end{align*}
- \begin{align*}(+2) \times (-4)\end{align*}
- \begin{align*}(+4) \times (-1)\end{align*}
Match each given phrase with the correct multiplication statement. Then, determine each product.
A. take away six groups of 3 balls
B. net worth after losing seven $5 bills
C. take away nine sets of 8 forks
D. take away four sets of four plates
E. receive eight groups of 4 glasses
F. buy seven sets of 12 placemats
a) \begin{align*}(+8) \times (+4)\end{align*}
b) \begin{align*}(+7) \times (-5)\end{align*}
c) \begin{align*}(-4) \times (+4)\end{align*}
d) \begin{align*}(-9) \times (+8)\end{align*}
e) \begin{align*}(+7) \times (+12)\end{align*}
f) \begin{align*}(-6) \times (+3)\end{align*}
Use the rules that you have learned for multiplying real numbers to answer the following problems.
1. \begin{align*}(-13) \times (-9)\end{align*}
2. \begin{align*}(-3.68) \times (82.4)\end{align*}
3. \begin{align*}\left(\frac{4}{9}\right) \times \left(\frac{5}{7}\right)\end{align*}
4. \begin{align*}\left(7 \frac{2}{3} \right) \times \left(6 \frac{1}{2}\right)\end{align*}
5. \begin{align*}(15.734) \times (-8.1)\end{align*}
What integer multiplication is modeled?
Answers
Use models of color counters...
- \begin{align*}(-7) \times (-2)\end{align*} This multiplication statement means to take away 7 sets of 2 yellow counters. \begin{align*}(-7) \times (-2)=14\end{align*}
- \begin{align*}(-5) \times (+3)\end{align*} This multiplication statement means to remove five groups of 3 red counters. \begin{align*}(-5) \times (+3)=-15\end{align*}
- \begin{align*}(+4) \times (-1)\end{align*} This multiplication statement means to add 4 groups of 1 yellow counter. \begin{align*}(+4) \times (-1)=-4\end{align*}
Match each given phrase...
A. take away six groups of 3 balls \begin{align*}\rightarrow\end{align*} f) \begin{align*}(-6) \times (+3)\end{align*}
B. net worth after losing seven $5 bills \begin{align*}\rightarrow\end{align*} b) \begin{align*}(+7) \times (-5)\end{align*}
C. take away nine sets of 8 forks \begin{align*}\rightarrow\end{align*} d) \begin{align*}(-9) \times (+8)\end{align*}
D. take away four sets of four plates \begin{align*}\rightarrow\end{align*} c) \begin{align*}(-4) \times (+4)\end{align*}
E. receive eight groups of 4 glasses \begin{align*}\rightarrow\end{align*} a) \begin{align*}(+8) \times (+4)\end{align*}
F. buy seven sets of 12 placemats \begin{align*}\rightarrow\end{align*} e) \begin{align*}(+7) \times (+12)\end{align*}
Use the rules...
- \begin{align*}(-13) \times (-9)=+117\end{align*}
\begin{align*}& \left(\frac{4}{9}\right) \times \left(\frac{5}{7}\right)\\ & \left(\frac{4 \times 5}{9 \times 7}\right)\\ &= \frac{20}{63}\end{align*}
- \begin{align*}(15.734) \times (-8.1)\end{align*} \begin{align*}& \ \ 15.734\\ & \underline{\times -8.1 \;\;\;}\\ & \quad \ \ 15734\\ & \underline{ \; \; \; 125872 {\color{blue}0}\;\;\;}\\ & -127.4454\end{align*}
What integer multiplication...
1. \begin{align*}(+4) \times (+3)\end{align*}
2. \begin{align*}(+5) \times (-3)\end{align*}
3. \begin{align*}(-3) \times (-2)\end{align*}
Summary
The multiplication of integers can be represented with manipulative such as color counters. Multiplication by a positive integer can be thought of as repeated addition and multiplication by a negative integer can be thought of as repeated subtraction. The examples presented in this lesson that were represented by models, could also have been represented on a number line. The results would have been the same. The rules for multiplying integers were shown by the models:
- The product of two positive integers is always positive.
- The product of two negative integers is always positive.
- The product of a positive integer and a negative integer is always negative.
These rules can be applied to the multiplication of all real numbers. In this lesson you also learned to multiply proper fractions as well as mixed numbers. The three steps that you applied to multiply fractions were:
- Multiply the numerators.
- Multiply the denominators.
- Simplify the fraction.
The final topic that was presented in this lesson was the multiplication of decimals. You learned that multiplying the decimal numbers was the same as multiplying whole numbers. The decimal point was inserted in the answer by counting from right to left, the number of places equal to the total number of digits after the decimal points in the problem.