36.1: Positive and Negative Numbers (2024)

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    Lesson

    Let's explore how we represent temperatures and elevations.

    Exercise \(\PageIndex{1}\): Notice and Wonder: Memphis and Bangor

    36.1: Positive and Negative Numbers (2)

    What do you notice? What do you wonder?

    Exercise \(\PageIndex{2}\): Above and Below Zero

    1. Here are three situations involving changes in temperature. Represent each change on the applet, and draw it on a number line. Then, answer the question.
      1. At noon, the temperature was 5 degrees Celsius. By late afternoon, it has risen 6 degrees Celsius. What was the temperature late in the afternoon?
      2. The temperature was 8 degrees Celsius at midnight. By dawn, it has dropped 12 degrees Celsius. What was the temperature at dawn?
      3. Water freezes at 0 degrees Celsius, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7 degrees Celsius. What is the freezing temperature of the gallon of salt water?
    2. Discuss with a partner:
      1. How did you name the resulting temperature in each situation? Did both of you refer to each resulting temperature by the same name or different names?
      2. What does it mean when the resulting temperature is above 0 on the number line? What does it mean when a temperature is below 0?
      3. Do numbers below 0 make sense outside of the context of temperature? If you think so, give some examples to show how they make sense. If you don’t think so, give some examples to show otherwise.

    Exercise \(\PageIndex{3}\): High Places, Low Places

    1. Here is a table that shows elevations of various cities.
      city elevation (feet)
      Harrisburg, PA \(320\)
      Bethell, IN \(1,211\)
      Denver, CO \(5,280\)
      Coachella, CA \(-22\)
      Death Valley, CA \(-282\)
      New York City, NY \(33\)
      Miami, FL \(0\)

      Table \(\PageIndex{1}\)
      1. On the list of cities, which city has the second highest elevation?
      2. How would you describe the elevation of Coachella, CA, in relation to sea level?
      3. How would you describe the elevation of Death Valley, CA, in relation to sea level?
      4. If you are standing on a beach right next to the ocean, what is your elevation?
      5. How would you describe the elevation of Miami, FL?
      6. A city has a higher elevation than Coachella, CA. Select all numbers that could represent the city’s elevation. Be prepared to explain your reasoning.
        1. -11 feet
        2. -35 feet
        3. 4 feet
        4. -8 feet
        5. 0 feet
    2. Here are two tables that show the elevations of highest points on land and lowest points in the ocean. Distances are measured from sea level. Drag the points marking the mountains and trenches to the vertical number line and answer the questions.
      point mountain continent elevation (meters)
      C Everest Asia \(8,848\)
      H Kilimanjaro Africa \(5,895\)
      E Denali North America \(6,168\)
      A Pikchu Pikchu South America \(5,664\)
      Table \(\PageIndex{2}\)
      point trench ocean elevation (meters)
      F Mariana Trench Pacific \(-11,033\)
      B Puerto Rico Trench Atlantic \(-8,600\)
      D Tonga Trench Pacific \(-10,882\)
      G Sunda Trench Indian \(-7,725\)
      Table \(\PageIndex{3}\)
      1. Which point in the ocean is the lowest in the world? What is its elevation?
      2. Which mountain is the highest in the world? What is its elevation?
      3. If you plot the elevations of the mountains and trenches on a vertical number line, what would 0 represent? What would points above 0 represent? What about points below 0?
      4. Which is farther from sea level: the deepest point in the ocean, or the top of the highest mountain in the world? Explain.

    Are you ready for more?

    A spider spins a web in the following way:

    • It starts at sea level.
    • It moves up one inch in the first minute.
    • It moves down two inches in the second minute.
    • It moves up three inches in the third minute.
    • It moves down four inches in the fourth minute.

    Assuming that the pattern continues, what will the spider’s elevation be after an hour has passed?

    Summary

    Positive numbers are numbers that are greater than 0. Negative numbers are numbers that are less than zero. The meaning of a negative number in a context depends on the meaning of zero in that context.

    For example, if we measure temperatures in degrees Celsius, then 0 degrees Celsius corresponds to the temperature at which water freezes.

    In this context, positive temperatures are warmer than the freezing point and negative temperatures are colder than the freezing point. A temperature of -6 degrees Celsius means that it is 6 degrees away from 0 and it is less than 0. This thermometer shows a temperature of -6 degrees Celsius.

    If the temperature rises a few degrees and gets very close to 0 degrees without reaching it, the temperature is still a negative number.

    36.1: Positive and Negative Numbers (3)

    Another example is elevation, which is a distance above or below sea level. An elevation of 0 refers to the sea level. Positive elevations are higher than sea level, and negative elevations are lower than sea level.

    36.1: Positive and Negative Numbers (4)

    Glossary Entries

    Definition: Negative Number

    A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

    36.1: Positive and Negative Numbers (5)

    Definition: Positive Number

    A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

    36.1: Positive and Negative Numbers (6)

    Practice

    Exercise \(\PageIndex{4}\)

    1. Is a temperature of -11 degrees warmer or colder than a temperature of -15 degrees?
    2. Is an elevation of -10 feet closer or farther from the surface of the ocean than an elevation of -8 feet?
    3. It was 8 degrees at nightfall. The temperature dropped 10 degrees by midnight. What was the temperature at midnight?
    4. A diver is 25 feet below sea level. After he swims up 15 feet toward the surface, what is his elevation?

    Exercise \(\PageIndex{5}\)

    1. A whale is at the surface of the ocean to breathe. What is the whale’s elevation?
    2. The whale swims down 300 feet to feed. What is the whale’s elevation now?
    3. The whale swims down 150 more feet more. What is the whale’s elevation now?
    4. Plot each of the three elevations as a point on a vertical number line. Label each point with its numeric value.

    Exercise \(\PageIndex{6}\)

    Explain how to calculate a number that is equal to \(\frac{2.1}{1.5}\).

    (From Unit 6.1.5)

    Exercise \(\PageIndex{7}\)

    Write an equation to represent each situation and then solve the equation.

    1. Andre drinks 15 ounces of water, which is \(\frac{3}{5}\) of a bottle. How much does the bottle hold? Use \(x\) for the number of ounces of water the bottle holds.
    2. A bottle holds 15 ounces of water. Jada drank 8.5 ounces of water. How many ounces of water are left in the bottle? Use \(y\) for the number of ounces of water left in the bottle.
    3. A bottle holds \(z\) ounces of water. A second bottle holds 16 ounces, which is \(\frac{8}{5}\) times as much water. How much does the first bottle hold?

    (From Unit 6.1.4)

    Exercise \(\PageIndex{8}\)

    A rectangle has an area of 24 square units and a side length of \(2\frac{3}{4}\) units. Find the other side length of the rectangle. Show your reasoning.

    (From Unit 4.4.2)

    36.1: Positive and Negative Numbers (2024)

    FAQs

    36.1: Positive and Negative Numbers? ›

    A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0. A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

    What are positive and negative numbers on a thermometer? ›

    On a thermometer, positive numbers represent temperatures above zero, and negative numbers represent temperatures below. When it's warmer than zero degrees, we use positive numbers. It is 15 degrees Celsius in the water.

    What numbers are positive and negative? ›

    Positive numbers are those which are greater than zero. Negative numbers are those which are less than zero.

    How do you calculate negative to positive? ›

    To get the sum of a negative and a positive number, use the sign of the larger number and subtract. For example: (–7) + 4 = –3.

    How do you find a number is positive or negative? ›

    If a number is greater than zero, it is a positive number. If a number is less than zero, it is a negative number. If a number equals to zero, it is zero.

    Does negative mean colder? ›

    For example, if we measure temperatures in degrees Celsius, then 0 degrees Celsius corresponds to the temperature at which water freezes. In this context, positive temperatures are warmer than the freezing point and negative temperatures are colder than the freezing point.

    Is negative temperature higher or lower? ›

    A substance with a negative temperature is not colder than absolute zero, but rather it is hotter than infinite temperature.

    Which number is positive? ›

    Standard numbers, anything greater than zero, are described as 'positive' numbers.

    Which number is positive number? ›

    A positive number is any number greater than zero. The definition of positive numbers include: Natural numbers: Natural numbers, also called counting numbers, such as 1, 2, 3, 4, 5, etc. are positive numbers.

    What are 3 positive numbers? ›

    1,2 and 3 are the positive integers and the result of their addition and their multiplication is same .

    How do you remember positive and negative rules? ›

    There are two simple rules to remember: When you multiply a negative number by a positive number then the product is always negative. When you multiply two negative numbers or two positive numbers then the product is always positive. 3 times 4 equals 12.

    Do 2 negatives make a positive? ›

    The fact that the product of two negatives is a positive is therefore related to the fact that the inverse of the inverse of a positive number is that positive number back again.

    How do you average positive and negative numbers? ›

    How to calculate the mean using negative numbers
    1. Find out how many negative numbers exist in the data set. A negative number is any value that's less than zero. ...
    2. Add up all positive numbers first. ...
    3. Calculate negative numbers. ...
    4. Divide by the number of figures in the data set.
    Feb 3, 2023

    How do you find the positive of a number? ›

    1. Multiply with Minus One to Convert a Positive Number. Unlike me, if you are good at maths, I am sure you know that when you multiply two minus signs with each other, the result is always positive. So you can use the same method in excel to convert a negative number into a positive.

    Is negative 18 cold? ›

    The fingers, nose, ears, toes, cheeks and chin are most vulnerable to frostbite, and should be covered as much as possible. When Is Cold Too Cold? Experts agree that it's best to stay indoors if the temperature falls below zero degrees Fahrenheit or the windchill dips below -18.

    Is negative 15 colder than negative 23? ›

    -23 °C is colder in a freezer than -15 °C. That is because with negative numbers, “higher” the number, lower the value.

    Is negative 20 too cold? ›

    A wind chill of -20° Fahrenheit (F) will cause frostbite in just 30 minutes. Frostbite causes a loss of feeling and a white or pale appearance in extremities, such as fingers, toes, ear lobes or the tip of the nose. If symptoms are detected, get medical help immediately!

    Is negative colder than positive? ›

    Negative absolute temperatures (or negative Kelvin temperatures) are hotter than all positive temperatures - even hotter than infinite temperature.

    Is negative Fahrenheit colder? ›

    Here is the answer, without further ado: neither. Neither of them is colder: -40° Fahrenheit is exactly equal to -40° Celsius!

    Is 7 positive or negative? ›

    Any number without a minus sign in front of it is considered to be a positive number, meaning a number that's greater than zero. So while -7 is negative seven, 7 is positive seven, or simply seven.

    Is positive number is greater than negative? ›

    No, negative numbers are all less than positive numbers.

    How do you explain negative numbers? ›

    In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset.

    Is positive 1 or 0? ›

    Positive numbers are greater than 0 and located to the right of 0 on a number line. Negative numbers are less than 0 and located to the left of 0 on a number line.

    Is 0 a positive or negative? ›

    Zero is neither positive nor negative. It is the only number with such characteristics. The numbers to the right of zero on the number line are positive and those on the left side are negative.

    Is 1 a positive or negative number? ›

    Positive numbers are any numbers greater than zero, for example: 1, 2.9, 3.14159, 40000, and 0.0005.

    Is negative 3 and positive 3 the same? ›

    – (-3) = 3 The opposite of negative three is a positive three. Absolute Value: Absolute Value of a number is the distance from zero to that number on the number line.

    Why do negatives cancel out? ›

    The negatives cancel out

    When you multiply two negative numbers together, though, the first one flips the sign from positive to negative, and the second one flips it back. In other words, negative numbers cancel each other out: when you multiply a negative number by a negative number, you get a positive number.

    What is an example of a negative number? ›

    Negative numbers are numbers that have a minus sign as a prefix. They can be integers, decimals, or fractions. For example, -4, -15, -4/5, -0.5 are termed as negative numbers.

    What is a positive times a negative? ›

    RULE 1: The product of a positive integer and a negative integer is negative.

    Does 3 negatives make a positive? ›

    Explanation: Recognize the rules of negative numbers: if two negative numbers are multiplied, the result is positive. However if three negative numbers are multiplied, the result is negative.

    How many positives for one negative? ›

    Negativity has a longer shelf life than positivity because of nature's elegant and hard-wired design of your nervous system. Scientists have discovered it takes three positive experiences to offset one negative experience.

    Is negative 3 bigger than negative 1? ›

    –1 is greater than –3, because –1 lies to the right of –3 on the number line. For less than you can use the <-sign.

    How to calculate difference between two numbers positive and negative? ›

    How to Subtract Positive and Negative Numbers
    • Change the minus sign (on the operator) to a plus sign. In this example, you're changing from subtraction to addition: ...
    • Change the number after that minus sign to its opposite. –60 + (–40) = ?
    • Add the numbers using the rules for adding positive and negative numbers.

    What is the positive number of 1 to 100? ›

    Here, we have to take the first 100 positive integers. So the series is: 1, 2, 3, 4… 98, 99, 100.

    How do I read my thermometer? ›

    Most thermometers have two scales for temperature, Fahrenheit and Celsius. Read the numbers for °F (degrees of Fahrenheit). Each long line is for 1°F temperature. The four shorter lines between each long line are for 0.2°F (two tenths) of a degree of temperature.

    Is Fahrenheit positive or negative? ›

    Here, \[{T_C}\] is the temperature in Celsius scale. Thus, we can see the Fahrenheit temperature can have negative values of temperatures.

    What number should a thermometer read? ›

    The average normal oral temperature is 98.6°F (37°C). A rectal temperature is 0.5°F (0.3°C) to 1°F (0.6°C) higher than an oral temperature.

    What is the order of negative numbers? ›

    How To Order Negative Numbers. When you order negative numbers, the larger the number after the negative sign, the smaller the number actually is. This means that -7 is smaller than -5, and -20 is smaller than -9. This can be quite confusing to children who are used to numbers getting larger as they increase in size.

    Is 91.0 a normal temperature? ›

    Hypothermia is a dangerously low body temperature, below 95° F (35° C). For context, normal body temperature is 98.6° F (37° C). Aside from creating serious discomfort, low body temperature—specifically, hypothermia—is considered a medical emergency.

    Is 96.98 A normal temperature? ›

    Official answer. There is no universally accepted normal body temperature – some medical textbooks state it as 36.8°C (98.24F), some as 37°C (98.6°F), and some just give a range such as 36.1°C to 37.2°C (96.98°F to 98.96°F).

    What does 35.7 body temp mean? ›

    Hypothermia is a dangerous drop in body temperature below 35C (normal body temperature is around 37C). It's a medical emergency that needs to be treated in hospital.

    How do you read a digital thermometer in Fahrenheit? ›

    Push the “°C/°F” button.

    Frontal scanning thermometers sometimes have a button specifically labeled “°C/°F.” Look on the front or the side of your thermometer to see if you have one. If so, use this button to freely change between Celsius and Fahrenheit.

    Is 37 a temperature? ›

    Your normal body temperature is approximately 37°C. A fever is usually when your body temperature is 37.8°C or higher. You may feel warm, cold or shivery. You can find out if you have a fever by using a thermometer to take your temperature.

    What should a digital thermometer read? ›

    A normal temperature measured in the mouth is 98.6°F (37°C) and can be between 97°F and 99°F (36.1°C to 37.2°C). Different people have slightly different normal body temperatures.

    Which temperature is positive? ›

    So the Kelvin scale of temperature is only positive.

    Which temperature is only positive? ›

    Celsius or Fahrenheit scale may have positive or negative values. But a Kelvin scale will have only positive values on the temperature scale.

    How do you write negative temperature? ›

    Negative Temperatures

    In Celsius and Fahrenheit, you can indicate negative temperatures with the words “negative” or “minus” (for temperatures written as words) or a minus sign (for numerals). For example: Temperatures were as low as negative thirty-four degrees Celsius. Temperatures were as low as –34°C.

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