Integers

Integers
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Negative Numbers

Numbers with negative sign before them are called negative numbers (or negative Integers).

↬ Idea of negative numbers evolved with the concept of debt.

Suppose, we had to return five marbles to our friend but we returned only three (two are lost somehow) then how many marble we got now, we would say 0 because we don’t have any but we are in debt of two marbles which mean when we got some we have to give two out of it to our friend. So right now we got −2 marbles.

↬ Negative also means opposite.

Like, loss is opposite to profit, positive and negative charges merely mean opposite to each other, minus temperature means temperature below 0 degree (standard temperature) on a temperature scale, etc.

So, negative numbers together with positive numbers can be used to describe the quantities having opposite values mathematically.

INTEGERS

“When we include negative of natural numbers to the collection of whole numbers we got new set of numbers called Integers.” Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}


↪ Integers are denoted by letter Z which stands for german word Zahlen which means numbers.
↪ In this collection, 1, 2, 3, … are said to be positive integers and -1, -2, -3, … are said to be negative integers.
Zero is defined as the integer which is neither negative nor positive.
↪ Concept of Integer can be easily understood using Number line.

Number Line

It serves the purpose of representing numbers graphically (or visually). It consists of a line on which numbers are laid at equal distance.

Integers on Number Line

➤ Mark points at equal distance on the line.
➤ Mark a point as zero ( usually the point in middle) on it.
➤ Points to right of 0 are positive integers and are marked +1, +2, +3, etc. or simply 1, 2, 3 etc.
➤ Points to left of 0 are negative integers and are marked -1, -2, -3, etc.
➤ The line continues left and right forever.

Order of Integers

➤ A number on left is less than number on right. e.g., -5 < -4, -2 < -1, -1 < 1, 3 < 4.

➤ A number on right is greater than number on left side. e.g., 5 > 4, 5 > 4, 1 > 2, 4 > 5 .
➤ On a number line the number increases as we move to the right.
➤ On a number line the number decreases as we move to the left.

Following points can be concluded from above discussion :

(a) Every positive integer is larger than every negative integer.

(b) Zero is less than every positive integer.

(c) Zero is larger than every negative integer.

(d) Zero is neither a negative integer nor a positive integer.

(e) Farther a number from zero on the right, larger is its value.

(f) Farther a number from zero on the left, smaller is its value.

Absolute Value

It means how far a number is from zero.

For example “3” is 3 away from zero, but “−3” is also 3 away from zero.
So the absolute value of 3 is 3, and the absolute value of −3 is also 3.

Addition of Integers

Addition on number line

sense of negative number \longleftarrow

sense of positive number \longrightarrow

4+5 = 9→4→5 = 9
4+(-5) = 4-5 = -1→4←5 = -1
-4+5 = 1←4→5 = 1
-4+(-5) = -4-5 = -9←4←5 = -9

Rules for addition of Integers :

  • when the integers have same sign
    • Add the numbers
  • keep the sign
  • 2+3 = 5, 3+6 =9, −4+(−3) =−7
  • when the the integers have different sign
    • Subtract the smaller number from the bigger number (bigger in respect of absolute value).
  • keep the sign of the bigger number.
  • 2−5 = −3, −3+7 = 4 , −6+4= −2

Additive Inverse

Additive inverse of an integer is the number we add to it to get 0.

Additive inverse of 5 is −5 as 5+(−5) = 0

Additive inverse of −3 is 3 as −3+(3) = 0

Additive inverse of a given number is its opposite or negative value.

Subtraction of Integers

Opposite (or negative) of positive number is its negative value.

e.g., −(4) = −4, −(7) = −7 ➝ Additive inverse 

Opposite (or negative) of negative number is its positive value.

e.g., −(−4) = +4, −(−7) = +7 ➝ Additive inverse

∴ Subtracting is same as adding additive inverse.

Subtraction on Number line

sense of negative number \longleftarrow
sense of positive number \longrightarrow
sense of negative of negative number\longrightarrow
sense of negative of positive number \longleftarrow

4-5 = -1→4←5 = -1
4-(-5) = 4+5=9→4→5 = +9
-4-5 = -9←4←5 = -9
-4-(-5) = -4+5= -1←4→5 = 1

Rules for subtraction of Integers :

  • Change the subtraction sign to addition sign
  • Change the sign of second number (opposite)
  • Follow the rule of addition
    • 2−3 = 2+(−3) = −1,
    • −4−3 = −4+(−3) =−7 ,
    • −3−(−6) = −3+(6) = 3,

Time to Think

State whether the following statements are correct or incorrect. Construct examples to check your answer :
(i) When two positive integers are added we get a positive integer.
(ii) When two negative integers are added we get a positive integer.
(iii) When a positive integer and a negative integer are added, we always get a negative integer.
(iv) Additive inverse of an integer 8 is (– 8) and additive inverse of (– 8) is 8.
(v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer.
(vi) (–10) + 3 = 10 – 3
(vii) 8 + (–7) – (– 4) = 8 + 7 – 4

Answer

(i) Correct
2+3=5,
4+7=11

(ii) Incorrect
-2 + (-5)= -7,
-21 + (-32)= -53

(iii) Incorrect, we get negative integer when the bigger integer is negative,
5 + (-3) = 2,
-7 + 5 = -2,
-12 + 17 = 5

(iv) Correct
(v) Correct

(vi) Incorrect , following is correct
(–10) + 3 = -10 + 3 = -7

(vii) Incorrect, following is correct
8 + (–7) – (– 4) = 8 – 7 + 4

⏪ HCF & LCMMultiplication | Division | Properties of Integers ⏩

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