• representing numbers in scientific notation:
• 1 million = 1 000 000 = 1.0 × 10⁶
• 1 millionth = 0.000 001 = 1.0 × 10⁻⁶

• 2.4 billion = 2 400 000 000 = 2.4 × 10⁹
• 2.4 billionths = 0.000 000 002 4 = 2.4 × 10⁻⁹

• 3.05 trillion = 3 050 000 000 000 = 3.05 × 10¹²
• 3.05 trillionths = 0.00 000 000 003 05 = 3.05 × 10⁻¹²
• comparing numbers:
  • identifying which number is greater:
    • 1.5 × 10³ is greater than 1.5 × 10² (comparing exponents)
    • 1.5 × 10³ is greater than 1.4 × 10³ (comparing the decimal values when the exponents are the same)
  • comparing numbers to zero:
    • 3.05 trillionth is very much closer to zero than 3.05 trillion, just as one tenth is closer to zero than 1 ten
  • comparing numbers absolutely:
    • 1 million is 999 999.999 999 greater than 1 millionth (i.e., 1 000 000 − 0.000 001 = 999 999.999 999)
  • comparing numbers relatively:
    • 1 million to 1 millionth = 1 000 000 : 0.000 001 = 1 000 000 000 000 : 1 (multiply both terms by 1 000 000), so 1 million is 1 trillion times 1 millionth

• The numerals 0 to 9 are used to form numbers. They are referred to as the digits in a number and each digit corresponds to a place value. A billion is “a thousand millions”, and a trillion is “a thousand billions” or “a million millions”. After the trillions period come quadrillions, quintillions, sextillions, septillions, octillions, and so on. Each period is 1000 times the preceding one.
• Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, in expanded notation, or in scientific notation. Large numbers may be expressed as a decimal number with the unit expressed in words. For example, 36.24 trillion is equivalent to 36 240 000 000 000 = 3.624 × 10¹³.
• When a number is expressed in scientific notation, there is only one non-zero digit to the left of the decimal point. Thus, 36.24 × 10¹² is not in scientific notation because there are two digits to the left of the decimal point. In scientific notation, 36 240 000 000 000 is written as 3.624 × 10¹³.
• In words, 37 020 005 205 is written and said as “thirty-seven billion twenty million five thousand two hundred five”. Sometimes an approximation to a large number is used to describe a quantity. For example, the number 37 020 005 205 may be rounded to 37 billion or 37.02 billion, depending on the amount of precision needed. 
• Understanding the magnitude of a large number may be done by comparing it to other numbers and quantities. For example:
  • One million seconds is around 11.5 days.
  • One billion seconds is around 32 years.
  • One trillion seconds is around 32 000 years.
• A number greater than 1 that is written in scientific notation can be written in standard notation by multiplying the decimal number by ten the number of times indicated by its exponent. For example, for 3.2 × 10⁵, 3.2 is multiplied by ten, five times. The result is 320 000.
• A number written in standard notation can be written in scientific notation. For a number greater than 1, a decimal point is positioned so that the first non-zero digit is to the left of the decimal point, and then the exponent for the base ten is determined by counting the number of times that decimal number needs to be multiplied by 10 to produce that number in standard notation. For example, 156 000 000 000 = 1.56 × 10¹¹.
• Very small numbers refer to numbers between 0 and 1. The closer the number is to zero the smaller the number is. These numbers can also be written in scientific notation. A negative exponent is used to indicate that the decimal number needs to be divided by 10 that many times. For example, for 5.2 × 10^-8, 5.2 is divided by 10 eight times to become 0.000000052. 
• To write a small number in scientific notation, the decimal point is positioned so that the first non-zero digit is to the left of the decimal point, and then the exponent is determined by counting the number of times that decimal number needs to be divided by 10 to produce that number in standard notation. For example, 0.0034 = 3.4 × 10^-3.
• Numbers expressed in scientific notation can be compared by considering the number of times the decimal number is multiplied or divided by ten. The more times it is multiplied by ten, the greater the number. The more times it is divided by ten, the smaller the number.

Note

• Every strand of mathematics relies on numbers.
• Some numbers have cultural significance.
• Real-life contexts can provide an understanding of the magnitude of large and small numbers.
• The number 1 in scientific notation is 1 × 10⁰.  
• The exponent on the base ten, in scientific notation, indicates the number of times the decimal number is multiplied or divided by ten, not how many zeros need to be included for a number to be written in standard notation.
• When inputting numbers electronically, the “^” sign is used for exponents; for example, 10⁶ would be entered as 10^6.

• R = real-number system, containing all of the rational and irrational numbers:
  • rational numbers:
    • 4, $$\frac{2}{3}$$, 0.5, 0, $$1.\overline{6}$$, −5, −$$\frac{9}{2}$$
  • irrational numbers:
    • π, $$\sqrt{2}$$, 4.515 515 551 555 515 555 51…
  • integers:
    • …, −3, −2, −1, 0, 1, 2, 3, …
  • whole numbers:
    • 0, 1, 2, 3, …
  • natural numbers:
    • 1, 2, 3, …

• Real numbers are a set of numbers that contain all rational and irrational numbers.
• Rational numbers are those that can be expressed in the form $$\frac{a}{b}$$, where a and b are integers; for example, −$$\frac{4}{3}$$, 3.12, $$\frac{1}{2}$$, −7, 0, 205, $$6.\dot4$$, −32.5.
• Fractions (positive and negative) are rational numbers. Any fraction can be expressed as a decimal number that either terminates or repeats. 
• Fractions can be written in a horizontal format (e.g., 1/2 or ½) as well as stacked format (e.g., $$\frac{1}{2}$$).
• Whole numbers are rational numbers since any whole number can be expressed as a fraction (e.g., 5 = $$\frac{5}{1}$$).
• Integers (whole numbers and their opposites) are rational numbers since any integer can be expressed as a fraction (e.g., −4 = −$$\frac{4}{1}$$, +8 = +$$\frac{8}{1}$$).
• Irrational numbers are numbers that cannot be expressed as a fraction. Examples of irrational numbers include decimal numbers that never repeat or terminate (e.g., 3.12122122212222…), pi (π), and square roots of non-perfect squares (e.g., $$\sqrt{2}$$).
• Rational and irrational numbers can be represented as points on a number line to show their relative distance from zero.
• The farther a number is to the right of zero on a horizontal number line, the greater the number.
• The farther a number is to the left of zero on a horizontal number line, the lesser the number.
• There are an infinite number of numbers in the real number system.

Note

• Since Grade 1, students have been working with whole numbers. The set of whole numbers (W) is a subset of integers (I), which are a subset of rational numbers (Q).
• Since Grade 1, students have been working with positive fractions, which are rational numbers. Negative fractions are introduced in Grades 7 and 8 as students represent, compare, and order negative fractions. Students will perform operations with negative fractions in secondary school because they are still developing the skills in Grade 8 to perform operations with integers.
• In Grade 7, students were introduced to pi, which is an irrational number. They may have worked with approximations of pi (3.14 or $$\frac{22}{7}$$), which are rational numbers. In Grade 8, students are introduced to other types of irrational numbers.
• In Grade 8, the focus is supporting students in making connections among the different number systems and the way they have been building knowledge through the years about the real number system.

Build a number line clothesline (see Grade 5: B1.2, Sample Task 1) between −10 and +10. Have students write on a card any integer, decimal, fraction (mixed number), or irrational number (e.g., square root or π) that falls between these two endpoints. Have them read the number on their card aloud, then place it on the number line. Ask them to highlight any equivalencies between decimals and fractions and explain the strategies they used to determine relative location. Have them think about where irrational numbers such as $$\sqrt{2}$$ and π (3.141 59…) should go. See B1.3 for work on approximating square roots.

• estimating a square root:
  • $$\sqrt{8}$$ is between $$\sqrt{4}$$ and $$\sqrt{9}$$
    • $$\sqrt{4}$$ = 2
    • $$\sqrt{9}$$ = 3
    • 2 < $$\sqrt{8}$$ < 3
    • Since $$\sqrt{8}$$ is closer to $$\sqrt{9}$$ than to $$\sqrt{4}$$, then $$\sqrt{8}$$ is closer to 3 than 2.

• calculating a square root (using a calculator):
  • $$\sqrt{8}$$ ≈ 2.83

• The inverse of squaring a number is to take its square root.
• Each positive number has two possible square roots. For example, the square roots of 9 are +3 and −3 because (+3)(+3) = 9 and (−3)(−3) = 9.
• The symbol $$\sqrt{  }$$ means the positive square root. The symbol ±$$\sqrt{  }$$ means both the positive and the negative square root.
• Depending on the context, only the positive square root may be appropriate. For example, given the area of a square, the length of its side is determined by taking the square root of the area. Since the side is a dimension, it makes sense to determine only the positive square root.
• Square roots of non-perfect squares are irrational and are left in radical form (e.g., $$\sqrt{3}$$) or approximated to a decimal number.
• Estimating the square roots of non-perfect squares involves identifying the two perfect squares that are closest to it. For example, $$\sqrt{60}$$ is between $$\sqrt{49}$$ and $$\sqrt{64}$$, so the first step is to determine the square root of those perfect squares, that is, $$\sqrt {49}$$ = 7 and $$\sqrt{64}$$ = 8. The next step is to estimate a value that is close to the closest square root. Since 60 is closer to 64 than to 49, then $$\sqrt{60}$$ can be estimated as 7.8.

Note

• Solving for a length using the Pythagorean theorem involves applying squares and square roots of numbers.
• A spatial interpretation of a square number is to think of the area of a square with side length the square root of the area (side × side or s²).
• If the area of a square is 9, its side length is $$\sqrt{9}$$ or 3.
  • 9 is a perfect square.
• If the area of a square is 5, its side length is $$\sqrt{5}$$.
  • 5 is an imperfect square and its square root is an irrational number, with a decimal that never repeats or terminates.

• Perfect squares can be calculated. Imperfect squares can only be estimated. Calculators give approximations of all square roots of non-perfect square numbers.

• using fractions, decimal numbers, or percents, depending on context:
  • finding 50% of an unknown number is the same as finding $$\frac{1}{2}$$ of it, or in decimal form, multiplying by 0.50
  • if something is 50% off, it might be easier to think of the cost as “half the full price”
  • if something is $$\frac{2}{5}$$ off, it might be easier to think of multiplying the price by 0.4 to get a rough estimate (determine 10% and then multiply that by 4)

• Converting between fractions, decimals, and percents often makes calculations and comparisons easier to understand and carry out.
• Fractions, decimals, and percents all describe relationships to a whole. While fractions may use any number as a denominator, decimal units are in powers of ten (tenths, hundredths, and so on) and percents express a rate out of 100 (“per cent” means “per hundred”).
• Relationships of quantities relative to a whole can be expressed as a fraction, a decimal number, and a percent. The choice of using a fraction, decimal number, or a percent can vary depending on the context of a problem.
• When fractions are considered as a quotient, the numerator is divided by a denominator and the result is a decimal representation that can be converted to a percent.
• To convert a percent to a fraction, it can first be represented out of 100 and then an equivalent proper fraction or mixed number can be made. For example, 104.6% = $$\frac{104.6}{100}$$ = $$\frac{1046}{1000}$$ = $$1 \frac{46}{1000}$$ = $$1 \frac{23}{500}$$.
• Some decimal numbers when converted to a percent result in whole number percents, and others result in decimal percents (e.g., 0.15 = 15%, 0.642 = 64.2%, 3.425 = 342.5%).
• Percents can be whole number percents (e.g, 32%, 168%) or decimal percents (0.5%, 43.6%, 108.75%). Percents can be understood as decimal hundredths.
• Percents can be composed from other percents. A 15% discount combines a 10% discount and a 5% discount. A 13% tax adds 10% and another 3% (3 × 1%).
• To convert a percent to a decimal number, the percent is divided by 100 (e.g., 35.4% = 0.354, 0.1% = 0.001).
• There are three types of problems that involve percents – determining the percent a quantity represents relative to a whole; finding the percent of a number; and finding a number given the percent.
• Common benchmark fractions, decimals, and percents include:
  • 150% = $$1 \frac{1}{2}$$ = 1.50
  • 100% = 1 = 1.00
  • 75% = $$\frac{3}{4}$$ = 0.75
  • 50% = $$\frac{1}{2}$$ = 0.50
  • 25% = $$\frac{1}{4}$$ = 0.25
  • 20% = $$\frac{1}{5}$$ = 0.20
  • 10% = $$\frac{1}{10}$$ = 0.10
  • 5% = $$\frac{1}{20}$$ = 0.05
  • 1% = $$\frac{1}{100}$$ = 0.01
  • 0.1% = $$\frac{1}{1000}$$ = 0.001
• Unit fraction conversions can be scaled to determine non-unit conversions. For example:
  • $$\frac{1}{4}$$ = 0.25 = 25%, so $$\frac{1}{8}$$ = 0.125 = 12.5% (half of one fourth).
  • $$\frac{1}{8}$$ = 0.125 = 12.5%, so $$\frac{3}{8}$$ = 0.375 = 37.5 (three times one eighth).

Note

• Sometimes when working with percents, students may work with complex fractions in which a decimal number is the numerator. An equivalent proper fraction or mixed number can be made by multiplying both the numerator and denominator by the appropriate number of tens.
• More than one strategy can be used to solve problems involving percents. For example, a coat is on sale for 25% off. The cost of the coat can be determined by finding 25% of the original price and then subtracting that discount value from the original price. Another strategy could be to determine 75% of the original price.