Twój koszyk jest pusty
Integers (revision)
Example 1
Judy measured the temperature in the morning of every school day during a week. On Monday, it was \latex{ 6 \;ºC }, while on Tuesday, it was \latex{ 3 \;ºC } higher. On Wednesday, it was \latex{ 4 \;ºC } lower than on Monday; on Thursday, it was \latex{ 6 \;ºC } lower, while on Friday, it was \latex{ 9 \;ºC } lower than on Monday.
What was the morning temperature on
- Tuesday;
- Wednesday;
- Thursday;
- Friday?
Solution
The temperatures:
- Tuesday: \latex{6 + 3 = 9\; (°C)};
- Wednesday: \latex{6 -4 = 2 \;(°C)};
- Thursday: \latex{6 -6 = 0\; (°C)};
- Friday: \latex{6 -9 = -3\; (°C)}.
The morning temperature on Tuesday was \latex{ 9 \;ºC }, on Wednesday \latex{ 2 \;ºC }, on Thursday \latex{ 0 \;ºC } and on Friday \latex{ –3 \;ºC }.
The difference of two natural numbers is
- a positive whole number if the minuend is greater than the subtrahend;
- zero if the minuend and the subtrahend are equal;
- a negative whole number if the minuend is smaller than the subtrahend.
Integers shown on a number line:
natural numbers
negative integers
positive integers
zero
...
\latex{-7}
\latex{-6}
\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{+7}
...
Integers
Natural
numbers
numbers
\latex{-20}
\latex{-5}
\latex{-3}
\latex{-4}
\latex{-2}
\latex{-1}
\latex{0}
\latex{7}
\latex{2}
\latex{5}
\latex{1}
The additive inverses of numbers
Numbers located at the same distance from zero on a number line and with different signs are the additive inverses of each other.
Example:
minus \latex{8}
plus \latex{8}
\latex{ –8 } and \latex{ 8 } are the additive inverses of each other
\latex{0}
\latex{-8}
\latex{+8}
With symbols:
\latex{-(+8) = -8};
the additive inverse of \latex{ a } is: \latex{ -a }
\latex{-(-8) = +8};
the additive inverse of \latex{ –a } is: \latex{ –(–a) = a }
\latex{-(-a) = a}
additive inverse of \latex{\large –a } is \latex{\large a }
The absolute value of numbers
The absolute value of a number is its distance from \latex{0} on the number line.
the absolute value of \latex{+8}
\latex{\mid+8\mid = +8 = 8};
the absolute value of \latex{0}
\latex{\mid0\mid = 0};
the absolute value of \latex{-8}
\latex{\mid-8\mid = +8 = 8}.
The absolute value of a positive number is the number itself.
The absolute value of zero is \latex{0}.
The absolute value of zero is \latex{0}.
The absolute value of a negative number is the additive inverse of the number.
Exercises
{{exercise_number}}. Write down the additive inverses of the following numbers.
- \latex{+9}
- \latex{-7}
- \latex{2,002}
- \latex{0}
- \latex{-200}
- \latex{-(+158)}
{{exercise_number}}. Write down the absolute values of the following numbers.
- \latex{-4}
- \latex{-7}
- \latex{-2,002}
- \latex{0}
- \latex{94}
- \latex{-(-7)}
- \latex{+50}
- \latex{-(+1)}
- \latex{2,000}
- \latex{570}
- \latex{-(+100)}
- \latex{-(-10)}
{{exercise_number}}. Without performing the subtractions, determine whether the following differences are positive, zero or negative.
- \latex{27-13}
- \latex{13-13}
- \latex{13-27}
- \latex{27-27}
- \latex{1,027-1,031}
- \latex{1,270-1,027}
- \latex{1,270-1,270}
- \latex{1,027-1,013}
- \latex{1,027-1,103}
- \latex{1,207-1,301}
- \latex{1,310-1,270}
- \latex{1,301-1,207}
- \latex{1,027-1,027}
- \latex{1,070-1,007}
- \latex{1,007-1,070}
- \latex{1,070-1,070}
{{exercise_number}}. Observe the following numbers.
\latex{1,999}; \latex{-5}; \latex{4}; \latex{+12}; \latex{-6}; \latex{0}; \latex{12}; \latex{+5}; \latex{-2,001}; \latex{-12}
Which one(s)
- are positive;
- are not negative;
- are smaller than \latex{ 0 };
- have a negative sign;
- are the additive inverses of each other;
- have equal absolute values;
- are equal;
- has the greatest absolute value;
- are located at equal distances from zero on the number line;
- have absolute values greater than the number itself;
- have additive inverses that are not greater than the number itself?
{{exercise_number}}. The result of which series of operations is positive?
- \latex{511 + 51 + 501}
- \latex{511-51-501}
- \latex{511 + 51-501}
- \latex{511\times51\times501}
- \latex{511-51 + 501}
- \latex{511 + 51\times501}
- \latex{511-51\times501}
- \latex{511\times51-501}
{{exercise_number}}. The result of which series of operations is negative?
- \latex{(496-524) + 28}
- \latex{496-(524 + 28)}
- \latex{496-524 + 28}
- \latex{496-524-28}
- \latex{496-(524-28)}
- \latex{496-524\times28}
- \latex{496\times524-28}
- \latex{ 496 × 51 × } the additive inverse of \latex{ 524 }
{{exercise_number}}. For which integer \latex{ a } is \latex{\mid a\mid \lt 5?} How many such numbers are there? Show the solutions on a number line.
{{exercise_number}}. Arrange the following numbers in ascending order. Show them on a number line.
\latex{-6}; \latex{-10}; \latex{+3}; \latex{\mid+4\mid}; \latex{-3}; \latex{-4}; \latex{\mid6\mid}; \latex{6}; \latex{-(-3)}; \latex{-(+4)}; \latex{\mid-6\mid}.
{{exercise_number}}. Decide whether the following statements are true or false. Justify your answers using a number line.
- Out of two numbers, the one with the smaller absolute value is the smaller number.
- Zero is greater than any non-negative number.
- Out of a negative and a positive number, the one with the greater absolute value is the smaller.
- The absolute value of \latex{ 0 } is smaller than the absolute value of any negative number.
- Out of a negative and a positive number, the additive inverse of the negative number is always greater than that of the positive number.
Quiz
Where are the numbers smaller than the absolute value of their additive inverses located on a number line?