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Absolute values and additive inverses

\latex{ -5 }

\latex{ -4 }

\latex{ -3 }

\latex{ -2 }

\latex{ -1 }

\latex{0 }

\latex{1 }

\latex{2 }

\latex{3 }

\latex{4 }

\latex{5 }

\latex{ -5 }

\latex{ -4 }

\latex{ -3 }

\latex{ -2 }

\latex{ -1 }

\latex{0 }

\latex{1 }

\latex{2 }

\latex{3 }

\latex{4 }

\latex{5 }

In the case of integers, there is always a number that is found at the same distance from zero on the number line as the number itself.

\latex{5}

\latex{4}

\latex{3}

\latex{2}

\latex{1}

\latex{0}

\latex{-1}

\latex{-2}

\latex{-3}

\latex{-4}

\latex{-5}

\latex{-6}

\latex{-5} and \latex{+5} are found at the same distance from \latex{ 0 }.
These numbers are called additive inverses.

The additive inverse of \latex{-5}:
\latex{-(-5) = +5}

The additive inverse of \latex{+5}:
\latex{-(+5) = -5}

Numbers with opposite signs found at the same distance from zero on a number line are called additive inverses. The additive inverse of zero is zero.

\latex{-5}

\latex{-4}

\latex{-3}

\latex{-2}

\latex{-1}

\latex{0}

\latex{1}

\latex{2}

\latex{3}

\latex{-5}

\latex{-4}

\latex{-3}

\latex{-2}

\latex{-1}

\latex{0}

\latex{1}

\latex{2}

\latex{3}

\latex{-5} is \latex{5} units away from \latex{0} on the number line.

\latex{3} is \latex{3} units away from \latex{0} on the number line.

The absolute value of a number equals its distance from \latex{ 0 } on the number line. Symbol \latex{|–5| = +5} (the absolute value of \latex{–5} is \latex{+5}).

Since distance cannot be negative, the absolute value of a number is always positive or \latex{0}.

The absolute value of a positive number is the number itself.
The absolute value of zero is \latex{ 0 }.
The absolute value of a negative number is its additive inverse.

Exercises

{{exercise_number}}. Below, you can see the freezing points of some materials. Which of the freezing points has the highest and lowest absolute value?

alcohol \latex{-114°C}

mercury \latex{-39°C}

diesel oil \latex{-30°C}

oxygen \latex{-219°C}

hydrogen \latex{-259°C}

water \latex{0°C}

{{exercise_number}}. Show the following numbers and their additive inverses on a number line.

a) \latex{-7}

b) \latex{+8}

c) \latex{+3}

d) \latex{-5}

e) \latex{-2}

f) \latex{0}

{{exercise_number}}. Write down the additive inverses of the following numbers (e.g. the additive inverse of \latex{-2} is \latex{-(-2) = +2}.)

a) \latex{-8}

b) \latex{+7}

c) \latex{+5}

d) \latex{-6}

e) \latex{+1}

f) \latex{-1}

g) \latex{0}

h) \latex{-45}

i) \latex{-75}

j) \latex{+78}

k) \latex{-128}

l) \latex{9,876}

{{exercise_number}}. Below, you can see the elevation of some places. Which elevation has the highest and lowest absolute value?

Debrecen \latex{121} \latex{ m }
the Nile's delta \latex{-13} \latex{ m }
Dead Sea \latex{-397} \latex{ m }
Pécs \latex{160} \latex{ m }
Caspian Depression \latex{-28} \latex{ m }
Szeged \latex{84} \latex{ m }

{{exercise_number}}. Place the following numbers on three separate number lines, then arrange them in descending order.

\latex{–} \latex{-2}; \latex{-7}; \latex{+8}; \latex{0}; \latex{-9}; \latex{+4}; \latex{+6} \latex{-4}
\latex{–} the additive inverses of the previous numbers
\latex{–} the absolute values of the previous numbers

What do you notice?

{{exercise_number}}. What is the absolute value of the following numbers? Write down the number with the lowest and highest absolute value.

a) \latex{-4}

b) \latex{-7}

c) \latex{+5}

d) \latex{0}

e) \latex{+2}

f) \latex{-183}

g) \latex{-12}

h) \latex{-23}

i) \latex{+15}

j) \latex{-17}

k) \latex{+1}

l) \latex{251}

m) \latex{-45}

n) \latex{+48}

o) \latex{-205}

p) \latex{+134}

r) \latex{-2,007}

s) \latex{-358}

{{exercise_number}}. List whole numbers with the following absolute values.

a) \latex{5}

b) \latex{8}

c) \latex{9}

d) \latex{28}

e) \latex{0}

f) \latex{+400}

g) \latex{-7}

h) \latex{10}

i) \latex{+12}

j) \latex{320}

k) \latex{-500}

l) \latex{100}

{{exercise_number}}. What is it equal to?

a) \latex{|-5|}

b) \latex{|-9|}

c) \latex{|0|}

d) \latex{|+27|}

e) \latex{-(+3)}

f) \latex{-(-3)}

g) \latex{-(-10)}

h) \latex{-(+540)}

i) \latex{-|-7|}

j) \latex{-|+43|}

k) \latex{-|0|}

l) \latex{-|-320|}

{{exercise_number}}. Place whole numbers on a number line whose additive inverses are

less than \latex{4};
greater than \latex{-2};
at most \latex{3};
at least \latex{5};
not less than \latex{2};
not greater than \latex{-5}.

A whole number

Help

greater than \latex{2}

less than \latex{2}

not greater
than \latex{2} (at most \latex{2})

not less
than \latex{2} (at least \latex{2})

\latex{2}

\latex{3}

\latex{4}

\latex{5}

\latex{1}

\latex{0}

\latex{-1}

\latex{0}

\latex{1}

\latex{2}

\latex{3}

\latex{4}

\latex{5}

\latex{-1}

\latex{1}

\latex{3}

\latex{5}

\latex{0}

\latex{2}

\latex{4}

\latex{-1}

\latex{0}

\latex{1}

\latex{2}

\latex{3}

\latex{4}

\latex{5}

{{exercise_number}}. What are the integers greater than \latex{ –10 } and less than \latex{ 10 } whose additive inverses are

a) \latex{-3};

b) greater than \latex{+6};

c) less than \latex{-3};

d) between \latex{-2} and \latex{-5};

e) at most \latex{+6} and at least \latex{+4};

f) greater than \latex{-3} and less than \latex{+2}?

{{exercise_number}}. Show on a number line single-digit integers whose absolute value is

a) less than \latex{3};

b) greater than \latex{5};

c) at most \latex{6};

d) at least \latex{4};

e) at least \latex{3} and at most \latex{5};

f) not greater than \latex{ 3 }.

{{exercise_number}}. Write the following numbers in the correct sets. (→)

\latex{|-7|}; \latex{-(-5)}; \latex{-(+8)}; \latex{-5}; \latex{|+2|}; \latex{+6}; \latex{|10|}; \latex{|0|}; \latex{-|+1|}; \latex{-(-3)}; \latex{|-12|}; \latex{|-3|}

Not less than \latex{+4}

Less than \latex{+7}

{{exercise_number}}. How many whole numbers have an absolute value of

three;
at most three;
at least three;
at most \latex{ 1,001 };
at least three and at most five;
at most \latex{ 6 } and at least \latex{ 3 }?

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

There is an integer whose absolute value is the number itself.
There are integers whose absolute values are not the numbers themselves.
There is no integer whose absolute value and additive inverse are equal.
There are three integers whose absolute values are equal.
The additive inverse of every non-positive integer is a positive number.

Quiz

The additive inverse of the absolute values of how many negative integers is greater than the current year?

Mathematics 5.

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