About positive and an adverse numbers The number lineAbsolute value of positive and an unfavorable numbers including positive and an unfavorable numbers Subtracting positive and negative numbers Multiplying optimistic and negative numbers separating positive and an unfavorable numbers CoordinatesComparing optimistic and an unfavorable numbers Reciprocals of an adverse numbers

### About optimistic and an unfavorable Numbers

Positive number are any kind of numbers greater than zero, for example: 1, 2.9, 3.14159, 40000, and 0.0005. Because that each hopeful number, there is a negative number the is its opposite. We create the the opposite of a optimistic number with a negative or minus sign in prior of the number, and also call this numbers an adverse numbers. The opposites the the number in the list over would be: -1, -2.9, -3.14159, -40000, and also -0.0005. An unfavorable numbers are much less than zero (see the number line for a much more complete explanation of this). Similarly, the contrary of any an adverse number is a hopeful number. For example, the contrary of -12.3 is 12.3.We carry out not consider zero to it is in a positive or negative number. The amount of any kind of number and its the opposite is 0.The *sign* that a number describes whether the number is hopeful or negative, because that example, the authorize of -3.2 is negative, and also the sign of 442 is positive.We may additionally write confident and negative numbers as fractions or combined numbers.

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The following fractions space all equal:

(-1)/3, 1/(-3), -(1/3) and also - 1/3.

The complying with mixed numbers space all equal:

-1 1/6, -(1 1/6), (-7)/6, 7/(-6), and - 7/6.

### The Number Line

The number line is a line labeled with positive and an unfavorable numbers in boosting order native left to right, the extends in both directions. The number line shown below is simply a tiny piece that the number heat from -4 come 4.

For any kind of two different places ~ above the number line, the number on the best is greater than the number ~ above the left.

Examples:

4 > -2, 1 > -0.5, -2 > -4, and 0 > -15

### Absolute worth of hopeful and an adverse Numbers

The variety of units a number is native zero ~ above the number line. The absolute worth of a number is always a confident number (or zero). Us specify the absolute value of a number *n* by creating *n* in between two upright bars: |*n*|.

Examples:

|6| = 6|-0.004| = 0.004|0| = 0|3.44| = 3.44|-3.44| = 3.44|-10000.9| = 10000.9

### Adding optimistic and an adverse Numbers

1) When adding numbers that the same sign, we include their absolute values, and also give the result the exact same sign.

Examples:

2 + 5.7 = 7.7(-7.3) + (-2.1) = -(7.3 + 2.1) = -9.4 (-100) + (-0.05) = -(100 + 0.05) = -100.05

2) When including numbers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and also give the result the sign of the number through the larger absolute value.

Example:

7 + (-3.4) = ?The absolute values of 7 and -3.4 space 7 and 3.4. Subtracting the smaller from the larger provides 7 - 3.4 = 3.6, and since the larger absolute worth was 7, we give the an outcome the same sign together 7, for this reason 7 + (-3.4) = 3.6.

Example:

8.5 + (-17) = ? The absolute values of 8.5 and -17 are 8.5 and 17. Subtracting the smaller from the larger provides 17 - 8.5 = 8.5, and also since the bigger absolute worth was 17, we offer the an outcome the very same sign together -17, so 8.5 + (-17) = -8.5.

Example:

-2.2 + 1.1 = ?The absolute values of -2.2 and also 1.1 space 2.2 and also 1.1. Individually the smaller sized from the larger gives 2.2 - 1.1 = 1.1, and since the larger absolute value was 2.2, we offer the result the very same sign as -2.2, for this reason -2.2 + 1.1 = -1.1.

Example:

6.93 + (-6.93) = ?The absolute worths of 6.93 and -6.93 are 6.93 and also 6.93. Subtracting the smaller sized from the larger gives 6.93 - 6.93 = 0. The sign in this case does not matter, because 0 and also -0 room the same. Note that 6.93 and -6.93 are opposite numbers. Every opposite numbers have this home that their amount is same to zero. 2 numbers that add up come zero are also called additive inverses.

### Subtracting hopeful and an adverse Numbers

Subtracting a number is the same as including its opposite.

Examples:

In the complying with examples, we convert the subtracted number to its opposite, and include the two numbers.7 - 4.4 = 7 + (-4.4) = 2.6 22.7 - (-5) = 22.7 + (5) = 27.7 -8.9 - 1.7 = -8.9 + (-1.7) = -10.6 -6 - (-100.6) = -6 + (100.6) = 94.6

Note the the result of subtracting two numbers have the right to be confident or negative, or 0.

### Multiplying positive and an unfavorable Numbers

To main point a pair of numbers if both numbers have the same sign, your product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the *opposite* that the product of your absolute worths (their product is negative). If one or both that the number is 0, the product is 0.

Examples:

In the product below, both numbers room positive, for this reason we just take your product.0.5 × 3 = 1.5

In the product below, both numbers room negative, so us take the product the their absolute values.(-1.1) × (-5) = |-1.1| × |-5| = 1.1 × 5 = 5.5

In the product that (-3) × 0.7, the very first number is negative and the second is positive, so us take the product of their absolute values, i m sorry is |-3| × |0.7| = 3 × 0.7 = 2.1, and also give this an outcome a an adverse sign: -2.1, therefore (-3) × 0.7 = -2.1

In the product of 21 × (-3.1), the first number is positive and the second is negative, so us take the product that their pure values, i m sorry is |21| × |-3.1| = 21 × 3.1 = 65.1, and give this result a an unfavorable sign: -65.1, therefore 21 × (-3.1) = -65.1.

**To multiply any number of numbers:**

1. Counting the number of negative numbers in the product. 2. Take it the product the their absolute values.3. If the number of negative numbers counted in action 1 is even, the product is just the product from action 2, if the number of negative numbers is odd, the product is the contrary of the product in action 2 (give the product in step 2 a an unfavorable sign). If any kind of of the number in the product is 0, the product is 0.

Example:

2 × (-1.1) × 5 (-1.2) × (-9) = ? Counting the number of an unfavorable numbers in the product, we check out that there space 3 an unfavorable numbers: -1.1, -1.2, and also -9. Next, we take the product the the absolute values of every number: 2 × |-1.1| × 5 × |-1.2| × |-9| = 2 × 1.1 × 5 × 1.2 × 9 = 118.8 Since there to be an odd number of numbers, the product is the contrary of 118.8, i m sorry is -118.8, for this reason 2 × (-1.1) × 5 (-1.2) × (-9) = -118.8.

### Dividing hopeful and an unfavorable Numbers

To division a pair of numbers if both numbers have actually the very same sign, divide the absolute worth of the very first number by the absolute value of the second number.To division a pair of number if both numbers have various signs, divide the absolute value of the an initial number by the absolute worth of the second number, and give this an outcome a negative sign.

Examples:

In the department below, both numbers room positive, therefore we simply divide as usual.7 ÷ 2 = 3.5

In the department below, both numbers space negative, therefore we divide the absolute value of the first by the absolute worth of the second. (-2.4) ÷ (-3) = |-2.4| ÷ |-3| = 2.4 ÷ 3 = 0.8

In the department (-1) ÷ 2.5, both number have different signs, therefore we divide the absolute value of the an initial number by the absolute value of the second, i beg your pardon is |-1| ÷ |2.5| = 1 ÷ 2.5 = 0.4, and give this result a negative sign: -0.4, therefore (-1) ÷ 2.5 = -0.4.

In the department 9.8 ÷ (-0.7), both number have different signs, for this reason we division the absolute value of the very first number through the absolute value of the second, which is |9.8| ÷ |-0.7| = 9.8 ÷ 0.7 = 14, and give this result a an adverse sign: -14, for this reason 9.8 ÷ (-0.7) = -14.

### Coordinates

Number coordinates are bag of numbers that are offered to determine points in a grid, loved one to a special point called the origin. The origin has coordinates (0,0). We can think the the beginning as the facility of the grid or the beginning point for finding all various other points. Any kind of other suggest in the grid has actually a pair of collaborates (x,y). The x value or x-coordinate speak how plenty of steps left or ideal the suggest is native the point (0,0), just like on the number heat (negative is left the the origin, hopeful is appropriate of the origin). The y value or y-coordinate speak how plenty of steps up or under the point is indigenous the point (0,0), (negative is under from the origin, confident is increase from the origin). Utilizing coordinates, we may offer the ar of any point in the grid we favor by merely using a pair the numbers.

Example:

The origin listed below is wherein the x-axis and the y-axis meet. Suggest A has collaborates (2.3,3), because it is 2.3 devices to the right and 3 units up native the origin. Point B has coordinates (-3,1), since it is 3 devices to the left, and 1 unit increase from the origin. Point C has coordinates (-4,-2.5), due to the fact that it is 4 units to the left, and 2.5 units down from the origin. Suggest D has collaborates (9.2,-8.4); the is 9 systems to the right, and 8.4 systems down native the origin. Suggest E has collaborates (-7,6.6); the is 7 devices to the left, and also 6.6 devices up from the origin. Allude F has works with (8,-5.7); that is 8 units to the right, and also 5.7 devices down indigenous the origin.

### Comparing optimistic and negative Numbers

We deserve to compare two various numbers by looking at their positions top top the number line. For any two various places ~ above the number line, the number ~ above the right is better than the number top top the left. Note that every hopeful number is greater than any an unfavorable number.

Examples:

9.1 > 4, 6 > -9.3, -2 > -8, and 0 > -5.5-2

### Reciprocals of negative Numbers

The mutual of a hopeful or negative fraction is derived by convert its numerator and also denominator, the sign of the new fraction remains the same. To find the reciprocal of a mixed number, first convert the mixed number come an not correct fraction, then switch the numerator and denominator the the wrong fraction. An alert that when you multiply an adverse fractions through their reciprocals, the product is always 1 (**NOT** -1).

Examples:

What is the reciprocal of -2/7? We just switch the numerator and denominator, and also keep the same sign: -7/2.

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What is the mutual of - 5 1/8? First, we transform to a negative improper fraction: -5 1/8 = - 41/8, then we switch the numerator and denominator, and keep the very same sign: - 8/41.