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Subtracting integers
\latex{(+8)-(+3)=(+5)}
\latex{(+8)-(-8)=(+16)}
\latex{(-2)-(-8)=(+6)}
Subtracting a positive number
Example 1
Perform the following subtractions.
a) \latex{(+3)-(+2)} \latex{\quad} b) \latex{(+3)-(+3)} \latex{\quad} c) \latex{(+3)-(+4)} \latex{\quad} d) \latex{(-1)-(+3)}
Solution
Illustrate the subtractions with cards used in the game The little banker.
a)
\latex{(+3)-(+2)=(+1)}
b)
\latex{(+3)-(+3)=0}
- \latex{4} money cards cannot be subtracted from \latex{3} money cards. Adding a debt card and a money card (zero in total) enables the subtraction of \latex{4} money cards.
\latex{(+3)-(+4)=(-1)}
- \latex{3} money cards cannot be subtracted from \latex{1} debt card. Adding \latex{3} money cards and \latex{3} debt cards (zero in total) enables the subtraction of \latex{3} money cards.
\latex{(-1)-(+3)=(-4)}
When subtracting a positive number, the difference will be smaller than the minuend.
Subtracting a negative number
Example 2
Perform the following subtractions.
a) \latex{(-5)-(-3)} \latex{\quad} b) \latex{(-5)-(-5)} \latex{\quad} c) \latex{(-5)-(-6)} \latex{\quad} d) \latex{(+2)-(-3)}
Solution
Illustrate the subtractions with the same cards.
In the first two cases, it is possible to subtract \latex{ 3 } and \latex{ 5 } debt cards respectively from \latex{ 5 } debt cards.
In the first two cases, it is possible to subtract \latex{ 3 } and \latex{ 5 } debt cards respectively from \latex{ 5 } debt cards.
a)
\latex{(-5)-(-3)=(-2)}
b)
\latex{(-5)-(-5)=0}
- \latex{ 6 } debt cards cannot be subtracted from \latex{ 5 } debt cards. Adding a debt card and a money card (zero in total) enables to subtract \latex{ 6 } debt cards.
\latex{(-5)-(-6)=(+1)}
- \latex{ 3 } debt cards cannot be subtracted from \latex{ 2 } money cards. Adding \latex{ 3 } debt cards and \latex{ 3 } money cards (zero in total) enables to subtract \latex{ 3 } debt cards.
\latex{(+2)-(-3)=(+5)}
When subtracting a negative number, the difference will be greater than the minuend.
Relationship between addition and subtraction
Example 3
Compare the following subtractions and additions.
a)
\latex{(+6)-(+2)}
\latex{(+6)+(-2)}
\latex{(+6)-(+2)}
\latex{(+6)+(-2)}
b)
\latex{(+2)-(-3)}
\latex{(+2)+(+3)}
\latex{(+2)-(-3)}
\latex{(+2)+(+3)}
c)
\latex{(-3)-(+2)}
\latex{(-3)+(-2)}
\latex{(-3)-(+2)}
\latex{(-3)+(-2)}
d)
\latex{(-4)-(-1)}
\latex{(-4)+(+1)}
\latex{(-4)-(-1)}
\latex{(-4)+(+1)}
Solution
Illustrate the operations with cards.
a)
\latex{(+6)-(+2)=(+4)}
\latex{(+6)+(-2)=(+4)}
b)
\latex{(+2)-(-3)=(+5)}
\latex{(+2)+(+3)=(+5)}
c)
\latex{(-4)-(-1)=(-3)}
\latex{(-4)+(+1)=(-3)}
d)
\latex{(-3)-(+2)=(-5)}
\latex{(-3)+(-2)=(-5)}
The results of the pairs of additions and subtractions are equal.
We can subtract any number by adding its additive inverse.
Exercises
{{exercise_number}}. In the third round of The little banker, the number of Zoe's debt cards decreased by \latex{ 20 }, while that of Pete's debt cards decreased by \latex{ 10 }. In the same round, the number of Ben's money cards decreased by \latex{ 15 }, while that of Brigitte's money cards decreased by \latex{ 13 }. Whose wealth increased and whose decreased? Who won the round and who lost the most amount of money?
{{exercise_number}}. Write down the corresponding operations and calculate how much money you have.
- You had €\latex{10} and spent €\latex{7}.
- You had €\latex{4} and spent €\latex{9}.
- You had a debt of €\latex{5} and spent €\latex{8}.
- You had a debt of €\latex{12} and your dad paid off €\latex{8} of it.
{{exercise_number}}. The thermometer shows \latex{ +5 }\latex{ °C }. What value will it show if the temperature decreases by
a) \latex{2}\latex{ °C };
b) \latex{5}\latex{ °C };
c) \latex{6}\latex{ °C };
d) \latex{8}\latex{ °C };
e) \latex{10}\latex{ °C };
f) \latex{15}\latex{ °C? }
{{exercise_number}}. What value did the thermometer originally show if, after an increase of \latex{ 6 } °C, it now shows
a) \latex{+2}\latex{ °C };
b) \latex{0}\latex{ °C };
c) \latex{-5}\latex{ °C };
d) \latex{-8}\latex{ °C };
e) \latex{-10}\latex{ °C };
f) \latex{-15}\latex{ °C? }
{{exercise_number}}. The temperature in an ice cave is \latex{ −2 }\latex{ °C }, while at the entrance of the cave, it is \latex{ +13 }\latex{ °C }. How much colder is it inside the cave than at the entrance?
{{exercise_number}}. What signs should be written in the \latex{\fcolorbox{black}{eaf1f9}{\phantom{a}}~} to make the equalities true?
a) \latex{(+5)-(+8)=(+5)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(-8)}
b) \latex{(+5)-(+2)=(+5)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(-2)}
c) \latex{(+7)-(-3)=(+7)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(+3)}
d) \latex{(+7)-(-9)=(+7)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(+9)}
e) \latex{(-6)-(-4)=(-6)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(+4)}
f) \latex{(-6)-(-8)=(-6)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(+8)}
g) \latex{(+3)-(+5)=(+3)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(-5)}
h) \latex{(-5)-(-8)=(-5)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(+8)}
i) \latex{(-2)-(+6)=(-2)~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(-6)}
j) \latex{0-(+3)=0~\fcolorbox{black}{eaf1f9}{\phantom{a}}~(-3)}
{{exercise_number}}. What signs should be written in the \latex{\fcolorbox{black}{eaf1f9}{\phantom{a}}~} to make the equalities true?
a) \latex{(+5)-(+8)=(+5)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~8)}
b) \latex{(+3)-(+6)=(+3)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~6)}
c) \latex{(-4)-(+2)=(-4)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~2)}
d) \latex{(-5)-(-9)=(-5)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~9)}
e) \latex{(+8)-(-3)=(+8)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~3)}
f) \latex{(+7)-(+9)=(+7)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~9)}
g) \latex{(-4)-(-8)=(-4)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~8)}
h) \latex{(-9)-(+3)=(-9)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~3)}
i) \latex{(+7)-(+4)=(+7)+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~4)}
j) \latex{0-(-5)=0+(\fcolorbox{black}{eaf1f9}{\phantom{a}}~5)}
{{exercise_number}}. Perform the subtractions.
a) \latex{(-7)-(+5)}
b) \latex{(-7)-(-5)}
c) \latex{(+7)-(+5)}
d) \latex{(+7)-(-5)}
e) \latex{(+5)-(+8)}
f) \latex{(-5)-(-8)}
g) \latex{(-5)-(+8)}
h) \latex{(+5)-(-8)}
i) \latex{(+9)-(+4)}
j) \latex{(-9)-(-4)}
k) \latex{(+9)-(-4)}
l) \latex{(-9)-(+4)}
{{exercise_number}}. Perform the subtractions and check your answers.
a) \latex{(+7)-(+3)}
b) \latex{(-7)-(-9)}
c) \latex{(+8)-(+4)}
d) \latex{(-9)-(+3)}
e) \latex{(-4)-(+8)}
f) \latex{(+3)-(-8)}
g) \latex{(-4)-(-3)}
h) \latex{(+5)-(-5)}
i) \latex{(-4)-(+2)}
j) \latex{(-8)-(-9)}
k) \latex{(+5)-(-6)}
l) \latex{(-1)-(+2)}
m) \latex{0-(-8)}
n) \latex{(-5)-0}
o) \latex{0-(+2)}
p) \latex{(-3)-(-8)}
{{exercise_number}}. What number should be subtracted from \latex{ -5 } to get the following results?
a) \latex{0}
b) \latex{-5}
c) \latex{-2}
d) \latex{+5}
e) \latex{+3}
f) \latex{-4}
g) \latex{+1}
h) \latex{-8}
{{exercise_number}}. What number should be subtracted from \latex{ +3 } to get the following results?
a) \latex{+3}
b) \latex{0}
c) \latex{-3}
d) \latex{+1}
e) \latex{-4}
f) \latex{+7}
g) \latex{-6}
h) \latex{+8}
{{exercise_number}}. Complete the table.
\latex{a}
\latex{b}
\latex{a-b}
\latex{-9}
\latex{-8}
\latex{+18}
\latex{+6}
\latex{-13}
\latex{-4}
\latex{-4}
\latex{+4}
\latex{+10}
\latex{-2}
\latex{-3}
\latex{+14}
\latex{+5}
\latex{-2}
\latex{-4}
\latex{+9}
{{exercise_number}}. The first Olympic Games were held in \latex{ 776 } BC in Olympia. How many \latex{years} ago did this happen?
{{exercise_number}}. What number should be subtracted from
a) \latex{+3} to get \latex{+7};
b) \latex{+8} to get \latex{-3};
c) \latex{-5} to get \latex{+7};
d) \latex{-7} to get \latex{-4};
e) \latex{-4} to get \latex{0};
f) \latex{-8} to get \latex{-8}?
{{exercise_number}}. On a winter day, the temperature on the highest peak of a mountain was \latex{ −23 }\latex{ °C }, while at the foot of the mountain, it was \latex{ −6 }\latex{ °C }. Where was it warmer and by how many \latex{degrees?}
{{exercise_number}}. What are the differences between the highest and lowest temperatures on each continent? Which continent has the greatest and the smallest temperature difference?
highest temperature
lowest temperature
Europe
Asia
Africa
North America
South America
Australia and Oceania
Antarctica
\latex{+50} \latex{°C}
\latex{+54} \latex{ °C }
\latex{+58} \latex{ °C }
\latex{+57} \latex{ °C }
\latex{+49} \latex{ °C }
\latex{+53} \latex{ °C }
\latex{+15} \latex{ °C }
\latex{-55} \latex{ °C }
\latex{-68} \latex{ °C }
\latex{-24} \latex{ °C }
\latex{-63} \latex{ °C }
\latex{-33} \latex{ °C }
\latex{-22} \latex{ °C }
\latex{-89} \latex{ °C }
{{exercise_number}}. Write down the next five members of each sequence.
a) \latex{15}; \latex{12}; \latex{9}; \latex{6}; ...
b) \latex{-4}; \latex{-6}; \latex{-8}; \latex{-10}; ...
c) \latex{-12}; \latex{-18}; \latex{-24}; \latex{-30}; ...
d) \latex{59}; \latex{47}; \latex{35}; \latex{23}; ...
e) \latex{15}; \latex{14}; \latex{12}; \latex{9}; \latex{5}; ...
f) \latex{10}; \latex{9}; \latex{7}; \latex{6}; \latex{4}; ...
{{exercise_number}}. Write word problems for the following subtractions.
a) \latex{(+5)-(+3)}
b) \latex{(+3)-(+8)}
c) \latex{(-3)-(-7)}
{{exercise_number}}. At which river kilometre will the boats anchor? Perform the operations on each boat to find it out.
\latex{(+20)+(-47)-(-36)}
\latex{(+20)+(-45)-(-32)}
\latex{(+20)+(-45)-(-38)}
\latex{(+20)+(-45)-(-34)}
\latex{(+20)+(-47)-(-32)}
{{exercise_number}}. Replace the boxes with numbers to make the relations true.
a) \latex{(-15)-(+7)-\fcolorbox{black}{eaf1f9}{\phantom{0000}}=0}
b) \latex{(+9)-(-17)+(-9)-\fcolorbox{black}{eaf1f9}{\phantom{0000}}=0}
c) \latex{(-23)-(-28)-\fcolorbox{black}{eaf1f9}{\phantom{0000}}\lt0}
d) \latex{(-32)+(-14)-\fcolorbox{black}{eaf1f9}{\phantom{0000}}\gt0}
Quiz
What are the results of the following calculations?
- \latex{\underbrace{(-1)+(-1)+(-1)+(-1)+...+(-1)}_{1,001\text{ times}}}
- \latex{\underbrace{(-1)-(-1)-(-1)-(-1)-...-(-1)}_{1,001\text{ times}}}
