There are many different ways to represent numbers, and we are going to explore four of them.

The first way to represent numbers is using words. When we say "I am six years old" or "let's go - it shouldn't take a billion years to get ready for dinner" we are using words to represent numbers. We are probably familiar with the numbers zero, one through twenty, and twenty-one through ninety-nine, but let's take a minute to review some other words that represent numbers.

If the definitions shown for billion, trillion, and quadrillion surprise you, you may live in a country where they are defined slightly differently than they are here. We are using the modern short scale, commonly used in English and Arabic speaking countries and in Brazil. Different countries and cultures don't just use different alphabets and eat different delicious foods for breakfast – they also use different words for some numbers, even within the English language!

We can combine these words to form numbers like "ten thousand" (one followed by 1 + 3 = 4 zeros), which is equal to 10,000, or "one hundred million" (one followed by 2 + 6 = 8 zeros), which is equal to 100,000,000. We can add in more words to say "two thousand and fifty-three", which is equal to 2,053, or "six million thirty thousand and twenty-one", which is equal to 6,030,021.

We can also use words to represent very small numbers. To make large numbers we kept multiplying by ten, adding a zero in front of our ones digit each time to make it larger and larger. To make small numbers we can keep dividing by ten, adding a zero behind our ones digit each time to make it smaller and smaller.

We can combine these words to form numbers like "one millionth" (one divided by a million), which is equal to 1/1,000,000 or 0.000001, or "one thousandth" (one divided by a thousand), which is equal to 1/1,000 or 0.001. We can again add in more words to say "two hundredths", which is equal to 2/100 or 0.02, or"twenty-six hundredths", which is equal to 26/100 or 0.26, or "thirty-seven thousandths" which is equal to 37/1,000 or 0.037.

The second way to represent numbers is with digits, using zero, the numbers one through nine, and a decimal point (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and .). We used digits to represent numbers in the middle column of the two tables above and in the examples in the text. These numbers might be so familiar to you that you didn't even notice we were doing this!

The third way to represent numbers is using scientific notation. This is a way to save time and eliminate errors by not writing out all of the zeros in a very large or a very small number. So far we've managed to write out numbers with digits, but imagine what would happen if we wanted to use a very large number, like the number of centimeters in a light-year. If I told you that there are roughly nine hundred and forty-six quadrillion centimeters (946,000,000,000,000,000) in a light-year, and asked you how many centimeters there were in 2.5 million (2,500,000) light-years, you might start to worry about keeping track of all those zeros!

With scientific notation, we can express each number by a value between one and ten followed by the number of times we need to multiply it by ten. Two point five million is 2.5 times six powers of ten, and nine hundred and forty-six quadrillion is 946 times fifteen powers of ten, or 9.46 times seventeen powers of ten. Knowing this, we can write these values in scientific notation.

The fourth way to represent numbers is using e notation (also called scientific e notation). This is very similar to using normal scientific notation. Instead of writing out a multiplication sign (×) and the number "10" with an exponent, we just write the letter "e" and follow it by the number of powers of ten that we need.

You may realize that e notation looks familiar, because this is the way that calculators implement scientific notation. (This is because most calculators cannot write a multiplication sign or an exponent with tiny digits placed higher than normal numbers.) In e notation, the "e" stands for exponent, and tells us the number of times to multiply by ten (or to divide by ten, if the exponent is a negative number).

When we measure things we determine the quantity, or how many of them there are. When you say "a marathon is more than twenty-six miles long" or "I bought two pounds of apples" or "the baby is three and a half months old" or "my pet giraffe would run more than thirty miles per hour for fresh apricots" we are using both numbers (like twenty-six, two, three and a half, and thirty) and units (like miles, pounds, months, and hours).

If we want to compare two different values or measurements, we need to write them both in the same units. Could you get a speeding ticket for driving 20 meters per second in a 45 miles per hour zone? If planet Mars is currently 200 million miles from Earth while Venus is 260 million kilometers away, which one is closer to us? Is the volume of the Hubble Space Telescope (four meters by four meters by thirteen meters) bigger or smaller than a school bus (nine feet by nine feet by thirty-five feet)? Let's see how to find out!

There are many different units for distances (Angstroms, centimeters, inches, feet, yards, meters, kilometers, miles, light-years, parsecs, kiloparsecs, megaparsecs), for time (milliseconds, seconds, minutes, hours, days, weeks, years, decades, centuries, megayears), and for weights (grams, ounces, pounds, kilograms, tons). We can combine distances and measure the sides of a two-dimensional region to determine an area (centimeters-squared, acres, square miles) or the sides of a three-dimensional shape to determine a volume (cubic inches, litres, gallons, cubic kiloparsecs). We can combine distance and time to form velocity (miles per hour, or kilometers per second) or acceleration (meters per second-squared).

When we measure a distance in one unit, for example, we can convert the value into any other unit of distance as long as we know the relative size of the two units. If one meter is equal to 100 centimeters (cm), then two meters is equal to 200 centimeters, while 3 centimeters is equal to 0.03 meters.

Let's examine some common units of distance. We'll look at imperial units (like inches, feet, yards, and miles) which are often used in daily life, metric units (like Ångstroms, centimeters, meters, and kilometers) which are used worldwide, and large units (astronomical units, light-years, parsecs, kiloparsecs, and megaparsecs) which are used to describe the vast distances between planets, stars and galaxies.

Old imperial units were defined in terms of typical parts of the human body, though they were eventually standardized, and are still used today. Science uses the metric system, defined by the meter (the distance that light travels in a vacuum in 1/299,792,458 of a second). When comparing the two systems, it can be helpful to remember that a yard is roughly 90% of a meter and a mile is about 160% of a kilometer.

One of the useful properties of the metric system is that it uses a set of prefixes to define all units in terms of the base unit: the meter. The Ångstrom, equal to 10^-10 meters, is named for the nineteenth century Swedish astrophysicist Anders Ångstrom who pioneered solar spectroscopy and showed that the Sun contains hydrogen. A micrometer, or micron, is 10^-6 meters, a millimeter is 10^-3 meters, a centimeter is 10^-2 meters, a kilometer is 10³ meters, and a megameter is 10⁶ meters. To convert from one such unit to another, we just multiply or divide by factors of ten.

At larger scales, we use a set of units inspired by the universe itself. The astronomical unit is the average distance between the Sun and the Earth (the radius of the Earth's orbit), and is used to define the distances betwen many objects in the solar system. The light-year, despite containing the word "year," is a measurement of distance but not time. It is simply the distance that light travels in a year. The nearest solar system to our own is located four light-years away from us. Its bigger sister the parsec is 3.26 light-years. (Some may recall that a parsec is the distance to an object with a parallax angle of one arcsecond when viewed from two positions separated by one astronomical unit.) Kiloparsecs and megaparsecs, like kilometers and megameters, just add factors of 10³ and 10⁶ to the base unit.

Let's practice converting the distance between the Earth and the Moon from units of feet to units of kilometers. We can first convert from feet to yards, then yards to meters, and then meters to kilometers, creating a daisy chain of unit conversions that shifts us step by step from one unit to another. We'll start with the fact that the average distance to the Moon is 1,261,392,000 feet.

Next we can review units of time.

The length of a second is defined by the frequency of a photon transitioning between two ground state energy levels within a Cesium atom. We are also familiar with a day being the amount of time between two sunsets, and a year being the amount of time it takes for the Earth to revolve once around the Sun.

How many seconds are in a week?

The metric units for mass are a measure of how much matter composes an object. In contrast, the weight of an object is a function of both its mass and the local gravitational field. Think of weight as how forcefully an object pushes down on a measuring scale, while mass is the number of particles it contains. On the surface of the Earth, there is an accepted equivalence between mass and weight. If you bring objects to the Moon, a smaller body whose lower mass exerts a weaker pull, however, the objects' masses will remain the same but their weights will be much smaller.

Let's calculate how many pounds there are in a kilogram here on Earth.

That was a lot of multiplying! For contrast, let's now try converting the speed of light from meters per second to kilometers per second.

Let's consider another number that may be familiar, the gravitational acceleration here on Earth, g (the downward pull we feel towards the Earth's center). This has units of distance per time-squared, and can be thought of as how quickly the speed of a falling object increases. Let's convert g from units of feet per minute-squared to meters per second-squared. Because time appears twice in the bottom of the units fraction, we will need to convert both time units from minutes to seconds, as well as dealing with our feet.

When we use scientific notation, we write numbers in a particular format. Each number is written as the product of a number equal to or greater than one and less than ten (the mantissa), and ten raised to an integer value (the exponent). Here are some examples of numbers written using scientific notation:

Notice that each mantissa is written out using a certain number of digits. For the third example it is clear why it is written to three significant figures (9 – 1 – 2); we need all three digits to create the value 9.12. For the first two examples, though, what do the trailing zeroes do? They tell us how accurate the mantissa is!

We can illustrate this by adding error bars to each value to tell us how much smaller or larger it could really be. When we make a measurement and write it down, we indicate to what accuracy we have made the measurement. Consider statements like "my glass is half full," or "let's have lunch around noon," or "my giraffe is 9'11.5" tall." The first two statements are rather vague, but the third statement implies that our measuring tape is good to the nearest tenth of an inch. We deduce that the giraffe is between 9'11.45" and 9'11.55" tall, because any number between these two values would round off to 9'11.5" when written using only one decimal place. We can rewrite our three examples similarly, as

When combining numbers written using significant figures, we report our results to match the least accurate number we used. When adding or subtracting numbers, this means that we write the end product out to the same number of decimal places as the least accurate input.

When multiplying or dividing numbers, this means that we write the end product out to the same number of significant figures as the least accurate input.