cadaVR anatomy by Mozaik Education

The multiplication of integers

Example 1

During a board game, Emy borrowed \latex{ 100 } \latex{ cents } three times, while Zoe borrowed \latex{ 200 } \latex{ cents } four times from the bank. The bank registers the loans by deducting the given amount from the player's account.

How much money did Emy and Zoe borrow altogether?
How much money do they have in their bank accounts if they did not have any money or debt before taking out the loans?

Solution

Emy borrowed \latex{ 100 \;cents } three times.

\latex{3\times(+100) = (+100) + (+100) + (+100) = +300 \,(cents)}.
Zoe borrowed \latex{ 200 \;cents } four times.
\latex{4\times(+200) = (+200) + (+200) + (+200) + (+200) = +800 \,(cents)}.

Emy borrowed \latex{ 300 \;cents }, while Zoe borrowed \latex{ 800 \;cents. }

Emy's balance:

\latex{3\times(-100) = (-100) + (-100) + (-100) = -300 \,(cents)}.
Zoe's balance:
\latex{4\times(-200) = (-200) + (-200) + (-200) + (-200) = -800 \,( cents)}.

Emy's balance is \latex{ –300 \;cents }, while Zoe's is \latex{ –800 \;cents }.

The sum of the same integers can also be expressed as a multiplication.

Example: \latex{\quad(+7) + (+7) + (+7) + (+7) + (+7) = 5\times(+7) = (+35);}

\latex{\quad(-7) + (-7) + (-7) + (-7) + (-7) = 5\times(-7) = (-35).}

Let's see how to multiply integers

Multiplication with two factors

One of the factors is a positive number, while the other varies.

\latex{(\textcolor{red}{+}3)\times(+4) = (\textcolor{red}{+}12)}

\latex{(\textcolor{red}{+}2)\times(+4) = (\textcolor{red}{+}8)}

\latex{(\textcolor{red}{+}1)\times(+4) = (\textcolor{red}{+}4)}

\latex{0\quad\times(+4)=0}

\latex{(\textcolor{009fe3}{-}1)\times(+4) = (\textcolor{009fe3}{-}4)}

\latex{(\textcolor{009fe3}{-}2)\times(+4) = (\textcolor{009fe3}{-}8)}

\latex{(\textcolor{009fe3}{-}3)\times(+4) = (\textcolor{009fe3}{-}12)}

\latex{4}

\latex{0}

\latex{-12}

\latex{-8}

\latex{-4}

\latex{0}

\latex{4}

\latex{8}

\latex{12}

If a positive number is

multiplied by a positive number, the product is positive;
multiplied by \latex{0}, the product is \latex{0};
multiplied by a negative number, the product is negative.

The absolute value of the product is the product of the absolute values
of the factors.

One of the factors is a negative number, while the other varies.

\latex{(\textcolor{red}{+}3)\times(-4) = (\textcolor{009fe3}{-}12)}

\latex{(\textcolor{red}{+}2)\times(-4) = (\textcolor{009fe3}{-}8)}

\latex{(\textcolor{red}{+}1)\times(-4) = (\textcolor{009fe3}{-}4)}

\latex{0\quad\times(-4)=0}

\latex{(\textcolor{009fe3}{-}1)\times(-4) = (\textcolor{red}{+}4)}

\latex{(\textcolor{009fe3}{-}2)\times(-4) = (\textcolor{red}{+}8)}

\latex{(\textcolor{009fe3}{-}3)\times(-4) = (\textcolor{red}{+}12)}

\latex{4}

\latex{0}

\latex{12}

\latex{8}

\latex{4}

\latex{0}

\latex{-4}

\latex{-8}

\latex{-12}

If a negative number is

multiplied by a positive number, the product is negative;
multiplied by \latex{0}, the product is \latex{0};
multiplied by a negative number, the product is positive.

The absolute value of the product is the product of the absolute values
of the factors.

So

\latex{(+3)\times(+4) = (+12) \text{ and } (-3)\times(-4) = (+12);}
\latex{(+3)\times(-4) = (-12) \text{ and } (-3)\times(+4) = (-12).}

The product of two numbers with the same signs is positive,
the product of two numbers with different signs is negative.

The absolute value of the product can be calculated by multiplying the absolute values of the factors.

Multiplication with several factors

Example 2

Calculate the products.

\latex{(-2)\times(-3)\times(-5)}
\latex{(+2)\times(-3)\times(+5)\times(-1)}
\latex{(-1)\times(-1)\times(-1)\times(-1)\times(-1)}

Solution

When there are more than two factors, multiply the first two factors, then multiply the resulting product by the next factor, and so on.

\latex{{\color{00a1e4}{\underbrace{\color{black}{\left(-2\right)\times\left(-3\right)}}_{+6}}}\times(-5) = (+6)\times(-5) = (-30)}
\latex{{\color{00a1e4}{\underbrace{\color{black}{\left(+2\right)\times\left(-3\right)}}_{-6}}}\times(+5)\times(-1)= {\color{00a1e4}{\underbrace{\color{black}{\left(-6\right)\times\left(+5\right)}}_{-30}}}\times(-1)=(-30)\times(-1)=(+30)}

Or you can rearrange and group the factors:

\latex{(+2)\times(-3)\times(+5)\times(-1) ={\color{00a1e4}{\underbrace{\color{black}{\left(+2\right)\times\left(+5\right)}}_{+10}}}\times{\color{00a1e4}{\underbrace{\color{black}{\left(-3\right)\times\left(-1\right)}}_{+3}}}={\color{00a1e4}{\underbrace{\color{black}{\left(+10\right)\times\left(+3\right)}}_{+30}}}}

\latex{(-1)\times(-1) = \textcolor{red}{(+1)}\quad \to \quad \text{\textcolor{red}{even} number } (-1)}

\latex{(-1)\times(-1)\times(-1) = \textcolor{00a1e4}{(-1)}\quad \to \quad \text{\textcolor{00a1e4}{odd} number } (-1)}
\latex{(-1)\times(-1)\times(-1)\times(-1) = \textcolor{red}{(+1)}\quad \to \quad \text{\textcolor{red}{even} number } (-1)}
\latex{(-1)\times(-1)\times(-1)\times(-1)\times(-1) = \textcolor{00a1e4}{(-1)}\quad \to \quad \text{\textcolor{00a1e4}{odd} number } (-1)}

If neither of the factors is zero, the product is

positive if the number of negative factors is even;
negative if the number of negative factors is odd.

The absolute value of the product can be calculated by multiplying the absolute values of the factors.

Exercises

Estimate the results before performing the operations.

{{exercise_number}}. Determine the rule and continue each sequence with \latex{ 3 } additional terms.

\latex{(+2); (-4); (+8); ...}

\latex{(-7); (+14); (-28); ...}

\latex{(-25); (-125); (-625); ...}

{{exercise_number}}. Arrange the results of the following multiplications in ascending order.

A: \latex{\fcolorbox{f6b900}{fef6e3}{$(-5)\times(+35)$}\qquad} B: \latex{\fcolorbox{f6b900}{fef6e3}{$(-5)\times(-2)$}\qquad} C: \latex{\fcolorbox{f6b900}{fef6e3}{$0\times(+35)$}\qquad} D: \latex{\fcolorbox{f6b900}{fef6e3}{$(-4)\times(-9)$}\qquad} E: \latex{\fcolorbox{f6b900}{fef6e3}{$(+12)\times(+3)$}}

{{exercise_number}}. What numbers should be written instead of the symbols, so that the products are

positive;

zero;

negative?

\latex{(+512)\times\qquad}

\latex{(-1,990)\times\qquad}

\latex{\times(-5,555)}

{{exercise_number}}. What is the product of the smallest positive two-digit number and the greatest positive two-digit number?

{{exercise_number}}. Which product is greater?

\latex{(-14)\times(-82)\times(+9)\times(-2) \text{\quad or\quad} (-3)\times(+41)\times(+6)\times(-28)}

\latex{(-26)\times(+4)\times(-93)\times(+4) \text{\quad or\quad} (+39)\times(-2)\times(+62)\times(+8)}

Compare the absolute values of the products.

{{exercise_number}}. What is the product?

\latex{(-46)\times(+96)\times(-10)}

\latex{(-69)\times(+64)\times(-5)\times(+2)}

\latex{(+138)\times(-32)\times(+10)\times(-1)}

\latex{(+10)\times(-48)\times(-92)}

{{exercise_number}}. The factors in a multiplication are three positive and four negative numbers.

What will the sign of the product be?
Change the sign of the factors to get a product with the same sign as the original and one with the opposite sign. How can you achieve this?

{{exercise_number}}. Perform the following multiplications.

\latex{(+17)\times(+19)\times(-23)}

\latex{(-2)\times(+3)\times(+5)\times(-7)\times(-11)\times(-13)}

\latex{(-101)\times(-103)\times(-107)}

\latex{12,345 679\times(-9)\times (+8)}

\latex{(-4)\times(+12,345 679)\times(-18)}

\latex{(-21)\times(-12,345 679)\times(-3)}

{{exercise_number}}. (\latex{ –5 }) times the product of which two numbers is

\latex{-180;}

\latex{+180;}

\latex{500;}

\latex{-500;}

\latex{0?}

Find several solutions.

{{exercise_number}}. How many times

\latex{(-3)\times(-3)\times(-3)} is
\latex{(-1)\times(-1)\times(-1)\times(-1)\times(-1)} is
\latex{(-2)\times(+3)\times(-5)} is

\latex{(-81);}
\latex{(-10);}
\latex{(-60)?}

{{exercise_number}}. Which of the following products are equal?

A

B

C

D

E

F

\latex{(-36)\times(-2)}

\latex{(-6)\times(+12)}

\latex{(+24)\times(-3)}

\latex{(+18)\times(+4)}

\latex{(-12)\times(+6)}

\latex{(-8)\times(-9)}

{{exercise_number}}. Replace the squares with numbers that make the equalities true.

\latex{(-3)\times\square = \square\times(-5) = (+12)\times(-15) = (-6)\times\square = (-60)\times\square}

\latex{\square\times(+8) = \square\times(-16) = (-48)\times(-4) = (-96)\times\square = (+48)\times\square}

\latex{(+32)\times\square = (-24)\times\square = (+8)\times(+12) = (-4)\times\square = (+16)\times\square}

{{exercise_number}}. Fill in the missing numbers.

\latex{-39}

\latex{-2}

{{exercise_number}}. The average of two numbers is \latex{ –34 }. One of the numbers is \latex{ –100 }. What is the other number?

{{exercise_number}}. Which number is multiplied by

\latex{-2} to get \latex{-50;}

\latex{+19} to get \latex{0;}

\latex{+24} to get \latex{-96?}

{{exercise_number}}. Form all possible pairs using the numbers \latex{ –10 }, \latex{ +17 }, and \latex{ –25 }, and multiply each pair. How much greater is the largest product than the smallest?

{{exercise_number}}. Decide whether the following statements are true or false.

If the product of two integers is positive, both numbers are positive.
If an integer is multiplied by its additive inverse, the product can be zero.
If the factors have the same sign, the product is a positive number.
If there is an odd number of negative factors, the product is negative.

Quiz

\latex{ 1848 } is a lucky \latex{year} because it has at least one \latex{day} when the product of the \latex{month} and the \latex{day} is the last two digits of the \latex{year.} Which of the following \latex{years} is not lucky?

\latex{{1995}}

\latex{{1996}}

\latex{{1997}}

\latex{{1998}}

\latex{{1999}}

Mathematics 6.

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