Positive and Negative Quantities

When we speak of a certain quantity of money, it may either be a gain or a loss, a receipt or a payment. Now it is quite clear that whilst a gain adds to our stock, a loss lessens it. Moreover gains and loss are so related that if we gain as much as we lose the effect on our stock is nothing. Hence a quantity of money forms a gain is said to be opposite in character to a quantity which forms a loss.

When we speak of distance measured from a point it may be in either two or opposite directions, either towards the north or towards the south of the point, either towards the east or towards the west, either towards to the north-east or south-west of the point and so on. It is also clear that distances measured towards the east are so related to those measured towards the west that if we first walk any distance towards the east and then towards the west there will be no change in our position with respect to the starting point. Hence a distance measured in any direction is said to be opposite in character to that measured in the opposite direction

New aspect to Plus & Minus Sign

It has been shown in the introduction how concrete quantities are represented by numbers. It now remains to be seen how quantities of the same class but of opposite character are distinguished in their numerical representation.

When we consider any pair of such quantities we prefix the sign ‘+’ before the numerical measures of one and the sign ‘-‘ before those of the other. It is quite immaterial which of the two quantities we select for representation by numbers preceded by the sign ‘+’, but when we have once made our choice, we must stick to it throughout any connected series of operations. The following principle will illustrate the following principle

Income and debt are evidently quantities of opposite character. If then we choose to represent incomes by numbers preceded by ‘+’, we must represent debts by numbers preceded by the sign ‘-‘, and vice versa.

Hence if in any problem we choose the sign ‘+’ for incomes and the sign ‘-‘ for debts +30 ,+40 ,+50, will respectively represent incomes of $30, $40, $50 whereas -30, -40, -50, will represent debts of $30, $40, $50. But if the contrary choice is made, then -10, -20, -30 will represent incomes and +10 +20 +30 will represent debts.

Note:

When no sign is prefixed to a number, then the ‘+’ sign is understood, thus $a$ and $+a$ have the same meaning.

Positive & Negative Quantities

Numbers or symbols proceeded by the sign ‘+’ or no sing are called positive quantities. Whilst those proceeded by the sign ‘-‘ are called negative quantities. Thus each of the expressions, 4, +6, $a$ , $+b$ are positive quantities and -4, -6, $-a$ , $-b$ are negative quantities.

Note:

In positive and negative quantities, quantity is used in the sense of numbers.
The absolute value of a positive or a negative quantity is its value considered a part of its sign. Thus if $a$ stands for 5 and $b$ stands for 3, $+ab$ and $-ab$ have the same absolute value, which is 15
It is important to bear in mind the meanings of such expressions as a gain of 20, a rise of -8 centimeters etc.

Addition

Definition: When two or more quantities are united together, the result is called their sum and the process of find the result is called addition.

When two positive quantities are added together, the sum is a positive quantity whose absolute value is equal to the arithmetical sum of the absolute values of those quantities.

When two negative quantities are added together, the sum is a negative quantity whose absolute value is equal to the arithmetical sum of the absolute values of those quantities.

The result when a negative quantity is added to a positive quantity:

If a positive and a negative quantity are added together, the sign of the result is positive or negative according as the absolute value of the negative quantity is less or greater than that of the positive quantity and the absolute value of the result is always equal to the difference between the absolute values of the quantities.

Since $a+(-b)= -(b-a)$ , when $b$ is greater than $a$ , putting $a=0$ we have $+(-b)=-b$ , i.e., to add a negative quantity is the same as subtract its absolute value and conversely, to subtract a positive quantity, is the same as add a negative quantity having the same absolute value.

When any number of quantities is added together, the result will be the same in whatever order the quantities may be taken.

Suppose a man starting from a place travels 6 kilometers to the north, and then travels back along the same path 8 kilometers to the south, and then his position at the end of the journey is 2 kilometers to the south of that place. Again if the same man travels 8 kilometers to the south and then travels 6 kilometers to the north, then also at the end of the journey he is still 2 kilometers to the south of that place.

Thus, we have $6+(-8)=(-8)+6$ each being equal to (-2) or, more briefly, we have $6-8=-8+6$ and have the same result in every other case,

Hence generally, $a-b=-a+b$

When a number of quantities are added together, they can be divided into groups and the result expressed as the sum of these groups.

When any number of quantities is to be added, some of which are positive and others are negative we collect the positive terms in one group and the negative terms in another group. We express the result as the sum of these two groups.

$3-7-8+6-4+2=(3+8+5)+(-7-9-6)$

To add two or more algebraic expressions write down the terms in succession with their proper signs. Thus, $a-b+c-d$ is the same as $a+(-b)+c+(-d)$ .

$a-2b+c-4d+e=a+(-2b)+c+(-4d)+e$

The ordinary rule for adding together compound expressions: Put the expressions under one another so that the different sets of like terms may stand in vertical columns and draw a line below the last expression, then add up each vertical column and put the result below it.

Example: Add together $3a-5b+7c-9d, -8c+5a-3d+7b, 4d+2c-a \: and \: 2b-3c+6d$

The 1st expression: $3a-5b+7c-9d$

The 2nd expression: $5a+7b-8c-3d$

The 3rd expression: $2b-3c+6d$

The 4th expression: $-a+2c+6d$

$\therefore$ sum $=7a+4b-2c-2d$

Examples

1. Find the value of $-a-bc-a^2b$ when $a=2, b=3, c=5$ .

$a=2, bc=3 \times 5=15, a^2b=2^2 \times 3=12$

Hence required value $=-2-15-12=-(2+15+12)=-29$

2. Find the value of $a-3b+2c-7d$ when $a=2, b=4, c=3 and d=1$

$a-3b+2c-7d \\ = 2+(-12)+6+(-7) \\ =-4+(-7) \\ = -11$

Subtraction

Definition: Any quantity, $b$ is said to be subtracted from any other quantity $a$ when a third quantity $c$ is found such that the sum of $b$ and $c$ is equal to $a$ . In other words, $c=a-b$ .

The quantity from which another quantity is subtracted is called the minuend and the quantity subtracted is called subtrahend. The result is called the difference or the remainder. Thus if $c=a-b$ , the minuend is $a$ , subtrahend is $b$ and difference or remainder is $c$ .

To subtract a positive quantity is the same as to add a negative quantity giving the same absolute value and to subtract a negative quantity is the same as to add a positive quantity having the same absolute value.

The ordinary rule for subtracting one compound expression from another:

Put the subtrahend below the minuend in such a way that the different sets of like terms may stand in vertical columns and a draw a line below the subtrahend, the supposing the sing of every term of the subtrahend is to be changed, write down the sum of each vertical column.

Removal and insertion of brackets

The laws for removal of brackets are:

If any number of terms be enclosed within a pair of brackets preceded by the sign ‘+’, the brackets may be struck out, as of no value.
If any number of terms be enclosed within a pair of brackets preceded by the sign ‘-‘, the brackets may be removed provided that the sign of every term within the brackets be changed, viz., ‘+’ to ‘-‘ and ‘-‘ to ‘+’.
The reason is obvious: Any expression, included within the brackets, proceeded by the sign ‘+’ has to be added to. On the other hand any expression enclosed within brackets preceded by the sign ‘-‘ has to be subtracted from what goes before. Thus, $a-b+(c-d+e)=a-b+c-d+e$ and $a-b-(c-d+e)=a-b-c+d-e$

The laws of insertion of brackets are:

Any number of terms in an expression may be enclosed within a pair of brackets with the sign ‘+’ prefixed.
Any number of terms in any expression may be enclosed within a pair of brackets with the sign ‘-‘ prefixed, if the sign of every term put within the brackets be altered.

Examples

1. Simplify: $[a-\left\{b-(c-d)\right\}]-[2a-\left\{3b+(2c-4d)\right\}]$

$a-\left\{b-(c-d)\right\}]=a-b+c-d \\ 2a-\left\{3b+(2c-4d)\right\}]=2a-3b-2c+4d$

Hence required expression $=[a-b+c-d]-[2a-3b-2c+4d]=-a+2b+3c-5d$

2. Find the value of $a-b+c$ when $a=5, b=-2, c=-3$

$a-b+c =5-(-2)=(-3) \\ =5+2-3=4$

3. Simplify: $2a-[3a+\left\{4b-(2a-b)+5a-7b\right\}]$

$2a-[3a+\left\{4b-(2a-b)+5a-7b\right\}] \\ =2a-[3a+\left\{4b-2a+b+5a\right\}-7b \\ =2a-[3a+\left\{5b+3a\right\}-7b] \\ =2a- [3a+5b+3a-7b] \\ =2a-6a+2b \\ =-4a+2b$

Exercise

Simplify: $2a-3b- (4a-6b) +(-2a+5b)$
Simplify: $\left\{2a-(3b-5c)\right\}-[a-\left\{2b-(c-4a)\right\}]$
Subtract $a-b+c$ from $3a+2b-c$
What is to be added to $x+2y+z$ to make it $z$
What is to be added to $a^4-2a^2b^2$ to make it $a^4+b^4$
Simplify: $2x+3y-z-3x-2y+z$
Add together: $a-2b+5c$ and $-7a+3b-8c$
If $a=5, b= 4, x=8, y =7$ find the numerical value of: $(3x^8+5y^5-20a^2+49b^8)+(17a^2-27b^8-23x^8)+(-y^5 +3b^8-3a^2)+(-23b^8-4y^5+7a^2 +20x^8)$
If 4 be the unit, what is meant by a gain of -25?
A man gains Rs 30 in one year, loses Rs 20 in the second year, loses Rs 40 in the third year and gains Rs 60 in the fourth year, how would you represent his gains in his successive year, taking Rs 2 as the unit?

New aspect to Plus & Minus Sign

Positive & Negative Quantities

Addition

Examples

Subtraction

Removal and insertion of brackets

Examples

Exercise

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