About confident and an adverse numbers The number lineAbsolute value of hopeful and an unfavorable numbers including positive and negative numbers Subtracting hopeful and an adverse numbers Multiplying positive and an adverse numbers splitting positive and an unfavorable numbers CoordinatesComparing positive and an adverse numbers Reciprocals of an unfavorable numbers

### About confident and an adverse Numbers

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

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The complying with fractions room all equal:

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

The complying with mixed numbers room all equal:

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

### The Number Line

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

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

Examples:

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

### Absolute worth of hopeful and an adverse Numbers

The number of units a number is from zero on the number line. The absolute worth of a number is constantly a hopeful number (or zero). Us specify the absolute value of a number *n* by creating *n* in 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 confident and an unfavorable Numbers

1) When including numbers the the exact same sign, we add their absolute values, and also give the an outcome the 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 adding numbers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the number through the bigger absolute value.

Example:

7 + (-3.4) = ?The absolute values of 7 and -3.4 are 7 and also 3.4. Individually the smaller from the larger gives 7 - 3.4 = 3.6, and also since the bigger absolute worth was 7, we provide the an outcome the very same sign as 7, so 7 + (-3.4) = 3.6.

Example:

8.5 + (-17) = ? The absolute values of 8.5 and also -17 space 8.5 and 17. Subtracting the smaller sized from the larger offers 17 - 8.5 = 8.5, and also since the bigger absolute worth was 17, we give the result the exact same sign together -17, so 8.5 + (-17) = -8.5.

Example:

-2.2 + 1.1 = ?The absolute worths of -2.2 and 1.1 space 2.2 and 1.1. Individually the smaller sized from the larger offers 2.2 - 1.1 = 1.1, and also since the larger absolute value was 2.2, we provide the result the same sign together -2.2, therefore -2.2 + 1.1 = -1.1.

Example:

6.93 + (-6.93) = ?The absolute values of 6.93 and also -6.93 space 6.93 and also 6.93. Individually the smaller from the larger gives 6.93 - 6.93 = 0. The authorize in this situation does no matter, since 0 and -0 room the same. Note that 6.93 and -6.93 space opposite numbers. Every opposite numbers have this building that their sum is same to zero. Two numbers that include up to zero are likewise called additive inverses.

### Subtracting optimistic and an unfavorable Numbers

Subtracting a number is the exact same as adding its opposite.

Examples:

In the adhering to examples, we transform the subtracted number to its opposite, and add 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 positive or negative, or 0.

### Multiplying hopeful and an unfavorable Numbers

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

Examples:

In the product below, both numbers are positive, so we just take their product.0.5 × 3 = 1.5

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

In the product the (-3) × 0.7, the first number is negative and the second is positive, so us take the product of their absolute values, which is |-3| × |0.7| = 3 × 0.7 = 2.1, and give this result a an adverse sign: -2.1, therefore (-3) × 0.7 = -2.1

In the product that 21 × (-3.1), the first number is positive and the second is negative, so we take the product the their absolute values, i beg your pardon is |21| × |-3.1| = 21 × 3.1 = 65.1, and also give this an outcome a negative sign: -65.1, therefore 21 × (-3.1) = -65.1.

**To main point any variety of numbers:**

1. Counting the number of an adverse numbers in the product. 2. Take the product of their absolute values.3. If the number of negative numbers counting in step 1 is even, the product is just the product from step 2, if the number of an adverse numbers is odd, the product is the contrary of the product in step 2 (give the product in step 2 a an adverse 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 adverse numbers in the product, we view that there space 3 negative numbers: -1.1, -1.2, and also -9. Next, us take the product the the absolute worths of every number: 2 × |-1.1| × 5 × |-1.2| × |-9| = 2 × 1.1 × 5 × 1.2 × 9 = 118.8 Since there were an odd variety of numbers, the product is opposing of 118.8, i m sorry is -118.8, therefore 2 × (-1.1) × 5 (-1.2) × (-9) = -118.8.

### Dividing hopeful and an unfavorable Numbers

To divide a pair of numbers if both numbers have actually the very same sign, division the absolute worth of the an initial number through the absolute value of the 2nd number.To division a pair of number if both numbers have various signs, division the absolute worth of the an initial number by the absolute value of the 2nd number, and give this an outcome a an unfavorable sign.

Examples:

In the department below, both numbers space positive, for this reason we just divide together usual.7 ÷ 2 = 3.5

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

In the division (-1) ÷ 2.5, both number have different signs, therefore we divide the absolute value of the an initial number by the absolute worth of the second, i m sorry is |-1| ÷ |2.5| = 1 ÷ 2.5 = 0.4, and give this an outcome a an unfavorable sign: -0.4, for this reason (-1) ÷ 2.5 = -0.4.

In the department 9.8 ÷ (-0.7), both number have different signs, so we division the absolute value of the very first number by the absolute worth of the second, which is |9.8| ÷ |-0.7| = 9.8 ÷ 0.7 = 14, and also give this an outcome a negative sign: -14, so 9.8 ÷ (-0.7) = -14.

### Coordinates

Number coordinates are pairs of numbers that are provided to identify points in a grid, family member to a special suggest called the origin. The beginning has works with (0,0). We have the right to think the the origin as the facility of the grid or the beginning point for finding all various other points. Any kind of other allude in the grid has actually a pair of works with (x,y). The x worth or x-coordinate tells how plenty of steps left or appropriate the point is from the suggest (0,0), similar to on the number line (negative is left that the origin, confident is best of the origin). The y value or y-coordinate tells how many steps up or down the point is from the allude (0,0), (negative is under from the origin, hopeful is increase from the origin). Making use of coordinates, we may give the place of any point in the network we choose by merely using a pair the numbers.

Example:

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

### Comparing positive and an unfavorable Numbers

We have the right to compare two different numbers by feather at your positions top top the number line. For any kind of two different places ~ above the number line, the number top top the best is better than the number top top the left. Keep in mind that every optimistic number is greater than any negative number.

Examples:

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

### Reciprocals of an unfavorable Numbers

The mutual of a positive or negative fraction is obtained by convert its numerator and denominator, the sign of the new portion remains the same. To find the reciprocal of a mixed number, first convert the blended number to an wrong fraction, then switch the numerator and denominator the the not correct fraction. An alert that when you multiply an adverse fractions with their reciprocals, the product is constantly 1 (**NOT** -1).

Examples:

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

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What is the reciprocal of - 5 1/8? First, we transform to a an adverse improper fraction: -5 1/8 = - 41/8, then us switch the numerator and also denominator, and also keep the same sign: - 8/41.