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Addition – pg.1/3

Posted in Math Theory by Lia on 11/17/2011

Addition: pg. 1, 23

Contents:

pg.1…..Addition  – about addition

pg.2…..Addition – exercises

pg.3…..Addition – answers to exercises

 

About addition

The result of addition is called the sum.

Note:

  • When a parenthesis that is preceded by a plus sign is removed, all the terms that were inside those parentheses  maintain their sign.

    Example: 7x + (3a – 5 + 4b – 2c) = 7x + 3a – 5 + 4b – 2c

  • When a parenthesis that is preceded by a minus sign is opened, all the terms that were inside those parentheses change their sign.

    Example: 5x – (4a – 3 + 2b – 5c) = 5x – 4a + 3 – 2b + 5c

a) If the numbers have same sign, add the absolute values of the numbers (see what the absolute value of a number is, on  Number Types) and the result will have the numbers’ sign.

Example:

  • The sum of: +3; +6 and +1 is: 3+6+1 = 10, and the sign of the result is a plus.

  • The sum of: -4; -5 and -3 is: (-4) + (-5) + (-3) = – 4 – 5 – 3 = – (4+5+3) = -12  and the sign of the result is a minus.

b) If the numbers have different signs there are more ways to add those numbers. We are going to show two ways of solving those numbers’ addition:

Example:

Add the numbers: – 12; + 3; – 6; – 1; – 2; and + 7 .This  means:

(-12)+3 + (-6) + (-1) + (-2)+7 = -12 + 3 – 6 – 1 – 2 + 7

 

  *  Addition of numbers with different signs. Method 1

When adding all numbers that have same sign, there will be two results.  One result will be a positive number equal with the sum of the positive numbers and the other result will be a negative number equal with the sum of the negative numbers.  Subtract the smaller absolute value of the two results (numbers) from the higher absolute value. Their result will have the sign of the higher absolute value number (see what absolute value of a number is, on  Number Types).

Example: -12 + 3 – 6 – 1 – 2 + 7=+(3 + 7) + (-12 – 6 – 1 – 2) = +10 + (- 21) = -11

In this case, first parenthesis has the positive numbers (+3 and +7) and the second has the negative numbers (-12, – 6, – 1 and – 2). The sum of the first parenthesis is positive 10 and its absolute value is 10. The sum of the second  parenthesis is negative 21 and its absolute value is 21. This means that 10 will be subtracted from 21, as 10 is smaller in absolute value. The result will be 11. Because the number, 21, has a higher absolute value, the sign of the result, 11, will have its sign which is a negative sign.

 

  *  Addition of numbers with different signs. Method 2

We can add the numbers in the sequence that they appear in the expression. We can add two consecutive numbers step by step, as shown below, until we end up with one number, the result.

-12+3-6-1-2+7  = (-12 +3) -6 -1 -2 +7 = (- 9) -6 – 1 -2 +7 = (- 9 -6) – 1 -2 +7 = (-15) -1 -2 + 7 = (-15 -1) -2 + 7 = (-16) -2 +7 = (-16 -2) +7 = -18 +7 = – 11

If you need to visualize this addition method, there is a sketch below (Fig. 2).

The sketch starts with drawing  the Number Line. Then, we proceed with placing a dot on the Number Line for the first number in our expression.  After that, we need to consider the value and the sign of the numbers. We place a new dot in a new location that is an many units as the value of the number, away from the previous location. If the sign of the number is minus, the new dot is places in the negative direction. If the sign of the number is plus, the new dot is places in the positive direction.

Addition, fig 2_2

Addition properties

  • Addition is Commutative, meaning that the order of the terms does not make any difference.

Example: a + b + c = b + a + c

  • Addition is Associative, meaning that the grouping does not make any difference.

Example: (a + b) + (c + d) = (a + c) + (b + d)

  • Identity element for addition is0. Any number added to 0  will retain it’s original value.

Example: a + 0 = a

  • An additive inverse is a number that, when added to a number, has as result 0.

Example: a – a = 0

In this example, the additive inverse to “a” is “– a.

 

Addition: pg. 1, 23

 

 

 

 

 

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