Adding Numbers With Different Signs

Adding Real Numbers

Learning Objective(south)

· Add 2 or more existent numbers with the same sign.

· Add two or more real numbers with different signs.

· Simplify past using the identity property of 0.

· Solve application problems requiring the add-on of real numbers.

Introduction

Calculation real numbers follows the same rules every bit adding integers. The number 0 has some special attributes that are very of import in algebra. Knowing how to add these numbers can exist helpful in real-world situations equally well every bit algebraic situations.

Rules for Adding Existent Numbers

The rules for adding integers apply to other real numbers, including rational numbers.

To add two numbers with the same sign (both positive or both negative)

· Add their absolute values.

· Give the sum the aforementioned sign.

To add ii numbers with dissimilar signs (one positive and one negative)

· Observe the divergence of their accented values. (Note that when you find the divergence of the absolute values, you always subtract the bottom accented value from the greater 1.)

· Give the sum the aforementioned sign equally the number with the greater accented value.

Remember—to add together fractions, you need them to accept the same denominator. This is still true when one or more of the fractions are negative.

Case
Trouble	Find
	and	This problem has three addends. Add the first ii, and then add the third. Since the signs of the starting time two are the same, find the sum of the absolute values of the fractions Since both addends are negative, the sum is negative.
	and	At present add the tertiary addend. The signs are different, so observe the divergence of their absolute values.
		Since , the sign of the final sum is the same equally the sign of .
Answer

Instance
Trouble	Find
	and	The signs are different, and then find the difference of their absolute values.
		Showtime rewrite equally an improper fraction, then rewrite the fraction using a common denominator. Now substitute the rewritten fraction in the problem.
		Decrease the numerators and go along the same denominator. Simplify to lowest terms, if possible.
Respond		Since , the sign of the final sum is the aforementioned as the sign of .

When you add decimals, remember to line upward the decimal points so you are adding tenths to tenths, hundredths to hundredths, and so on.

Example
Problem	Discover 27.832 + (−iii.06).
	Since the addends have unlike signs, subtract their absolute values. \|−three.06\| = 3.06 The sum has the same sign every bit 27.832 whose absolute value is greater.
Answer	27.832 + (−3.06) = 24.772

Notice − 32.22 + 124.iii.

A) 19.79

B) 44.65

C) 92.08

D) 156.52

Show/Hide Answer

A) 19.79

Wrong. To find the sum of numbers with different signs, you lot must subtract their absolute values. When adding and subtracting decimals, you must pay attention to identify value. You may have subtracted 32.22 – 12.43 instead of 124.iii – 32.22. The correct answer is 92.08.

B) 44.65

Wrong. To discover the sum of numbers with unlike signs, y'all must decrease their absolute values. You may have added instead. Also, when adding and subtracting decimals, you must pay attention to place value. The right answer is 92.08.

C) 92.08

Right. Since the addends take different signs, you must subtract their absolute values.

124.three – 32.22 is 92.08. Since |124.3| > | − 32.22|, the sum is positive.

D) 156.52

Incorrect. To find the sum of numbers with different signs, y'all must subtract their accented values. Yous added their accented values. The correct respond is 92.08.

The Condiment Identity

The rules for calculation real numbers refer to the addends being positive or negative. But 0 is neither positive nor negative.

It should be no surprise that you add 0 the way you ever accept—adding 0 doesn't modify the value.

7 + 0 = 7	− vii + 0 = − vii
0 + 3.half-dozen = three.vi
ten + 0 = 10	0 + x = x

Notice that x + 0 = x and 0 + x = ten. This ways that information technology doesn't affair which addend comes start.

The number 0 is called the additive identity. The identity belongings of 0 states that adding 0 to other numbers doesn't alter their value. Yous can think of it in this fashion: calculation 0 lets the other number keep its identity.

What is 0 + y, when y = 3?

A) − three

B) 0

C) 3

Prove/Hibernate Respond

A) − 3

Incorrect. The identity property says 0 + any number = that number. Substituting 3 for y gives 0 + 3, and 0 + 3 = 3.

B) 0

Wrong. The sum of a number and 0 is that number, not 0. Substituting three for y gives 0 + 3, and 0 + three = 3.

C) 3

Correct. Substituting 3 for y gives 0 + three, and 0 + 3 = 3.

Applications of Improver

In that location are many situations that use negative numbers. For example, temperatures colder than 0° are normally described using negative numbers. In golf tournaments, players' scores are oft reported as a number over or under par, instead of the total number of strokes it takes to hit the brawl into the hole. (Par is the expected number of strokes needed to complete a hole.) A number nether par is negative, and a number over par is positive.

The following examples show how add-on of real numbers, including negative numbers, can be useful.

Example
Problem	Boston is, on boilerplate, 7 degrees warmer than Bangor, Maine. The low temperature on one cold winter twenty-four hour period in Bangor was −thirteen° F. Near what low temperature would you expect Boston to have on that day?
	If the temperature in Bangor is x, the temperature in Boston is x + 7.	The phrase "7 degrees warmer" means you add vii degrees to Bangor's temperature to estimate Boston's temperature.
	10 = −13	On that twenty-four hours, Bangor's low was −13°.
	Boston'south temperature is −xiii + 7 − 13 + 7 = −6	Substitute −thirteen for x to get Boston'due south temperature. Add the integers. Since one is positive and the other is negative, y'all observe the deviation of \|−xiii\| and \|seven\|, which is 6. Since \|−13\| > \|7\|, the last sum is negative.
Respond	You would expect Boston to have a temperature of −half dozen degrees.

Case
Problem	Before Joanne could deposit her paycheck of $802.83, she overdrew her checking business relationship. The residual was − $201.35. What was her balance later on she deposited the paycheck?
	− 201.35 + 802.83 − 201.35 + 802.83 = 601.83	By depositing her paycheck, Joanne is adding money to her business relationship. The new balance is the sum of the old (−201.35) and the paycheck amount. Since the numbers take different signs, find the divergence of −201.35. Since \|802.83\| > \|−201.35\|, the sum is positive.
Answer	The new balance is $601.48.

When forces or objects are working in reverse directions, sometimes it'due south helpful to assign a negative value to one and a positive value to the other. This is done often in physics and engineering, but it could also exist done in other contexts, such as football or a tug-of-war.

Example
Trouble	2 people are in a tug-of-state of war competition. They are facing each other, each holding the cease of a rope. They both pull on the rope, trying to move the center toward themselves. Here's an illustration of this situation. The person on the correct is pulling in the positive direction, and the person on the left is pulling in the negative management. At one point in the competition, the person on the right was pulling with 122.8 pounds of force. The person on the left was pulling with 131.three pounds of force. The forces on the middle of the rope, and then, were 122.viii lbs and − 131.three lbs. a) What was the net (total sum) force on the eye of the rope? b) In which direction was information technology moving?
	Cyberspace force = 122.8 +( − 131.3)	The internet strength is the sum of the 2 forces on the rope.
	Net force = − 8.5	To find the sum, add the divergence of the absolute values of the addends. Since \| − 131.iii\| > 122.viii, the sum is negative.
Answer	The net strength is −8.5 lbs (or viii.5 lbs to the left). The heart of the rope is moving to the left (the negative management).	Detect that it makes sense that the rope was moving to the left, since that person was pulling with more force.

After Bangor reached a low temperature of − thirteen°, the temperature rose but 4 degrees higher for the balance of the day. What was the high temperature that day?

A) − 17

B) − 9

C) 9

D) 17

Bear witness/Hide Respond

A) − 17

Incorrect. Although y'all used the correct sign, you found the sum of the absolute values. The temperature rose (added) 4 degrees from − 13, so the loftier temperature is − 13 + 4. Since the addends have unlike signs, yous must find the difference of the absolute values. | − 13| = xiii and |4| = 4. The divergence is 13 – four = 9. The sign of the sum is the aforementioned as the addend with the greater absolute value. Since | − 13| > |4|, the sum is negative. The correct answer is − 9.

B) − 9

Correct. The temperature rose (added) 4 degrees from − 13, then the loftier temperature is − 13 + four. Since the addends accept different signs, you must find the deviation of the accented values. | − 13| = 13 and |iv| = iv. The departure is xiii – 4 = 9. The sign of the sum is the aforementioned as the addend with the greater absolute value. Since | − 13| > |4|, the sum is − 9.

C) ix

Incorrect. Y'all found the difference of the absolute values of the addends. Yet, because | − 13| > |four|, the sign of the sum should be the same every bit the sign on − 13. The correct answer is − 9.

D) 17

Wrong. You added the absolute values of the addends, and gave the sum the wrong sign. The temperature rose (added) 4 degrees from − 13, so the loftier temperature is − 13 + 4. Since the addends have different signs, you lot must discover the difference of the absolute values. | − 13| = xiii and |4| = 4. The departure is thirteen – four = 9. The sign of the sum is the aforementioned equally the addend with the greater absolute value. Since | − 13| > |4|, the sum is negative. The correct answer is − 9.

Summary

As with integers, adding real numbers is done post-obit two rules. When the signs are the same, you add the absolute values of the addends and use the same sign. When the signs are different, you decrease the accented values and use the aforementioned sign equally the addend with the greater accented value. Adding 0, which is neither positive nor negative, is done using the additive identity of 0: x + 0 = x and 0 + x = ten, for any value of x.