• 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. For example, in the number 876 345 107, the digit 8 represents 8 hundred millions, the digit 7 represents 7 ten millions, the digit 6 represents 6 millions, the digit 3 represents 3 hundred thousands, the digit 4 represents 4 ten thousands, the digit 5 represents 5 thousands, the digit 1 represents 1 hundred, the digit 0 represents 0 tens, and the digit 7 represents 7 ones.
• Reading numbers involves interpreting them as a quantity when they are expressed in words, in standard notation, or in expanded notation using powers of ten. Large numbers may be expressed as a decimal number with the unit expressed in words. For example, 36.2 million is equivalent to 36 200 000 = 36.2 × 10⁶.
• Expanded notation with powers of ten shows a number as an expression by using addition, multiplication, and exponents. The number “three hundred seven million, twenty thousand, and fifty”, can be expressed as 307 020 050 and 3 × 10⁸ + 7 × 10⁶ + 2 × 10⁴ + 5 × 10.
• Numbers without units identified are assumed to be based on ones.
• Numbers can be written in terms of another number. For example:
  • 1 billion is 1000 millions.
  •  1 billion millimetres is equal to 1000 kilometres.
  • 1 billion seconds is about 32 years.
• Sometimes an approximation of a large number is used to describe a quantity. For example, the number 7 238 025 may be rounded to 7 million, or 7.2 million or 7.24 million, depending on the amount of precision needed.
• Numbers can be compared by their place value or they can be compared using proportional reasoning. For example, 1 billion is 1000 times greater than 1 million.

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 numbers. For example:
  • 1 billion seconds is about 32 years.
  • When they are closest to each other, Earth and Saturn are 1.2 billion kilometres apart.
  • Given that Earth’s population is 7.5 billion (and counting), if you are “one in a million”, there are 7500 people just like you.   
• There are patterns in the place value system that help people read, write, say, and understand numbers, and that suggest important ways for numbers to be composed and decomposed.
  • The place (or position) of a digit determines its value (place value). The 5 in 511, for example, has a value of 500, not 5.
  • A zero in a column indicates that there are no groups of that size in the number. It serves as a placeholder, holding the other digits in their correct “place”.
  • The value of the columns increases and decreases by powers of ten. With each shift to the left, a digit’s value increases by a power of 10 (i.e., its value is ten times as great). With each shift to the right, a digit’s value decreases by a power of 10 (i.e., its value is one tenth as great). 
  • To find the value of a digit in a number, the value of a digit is multiplied by the value of its place.
  • Each period – thousands, millions, billions, trillions – is 1000 times as great as the previous period. Periods increase in powers of 1000 (10³).

• When inputting numbers electronically, the “^” sign is used for exponents; for example, 10⁶ would be entered as 10^6.

• Any whole number multiplied by itself produces a square number, or a perfect square, and can be represented as a power with an exponent of 2. For example, 9 is a square number because 3 × 3 = 9, or 3².
• The inverse of squaring a number is to take its square root. The square root of 9 () is 3.
• A perfect square can be represented as a square with an area equal to the value of the perfect square. The side length of a perfect square is the square root of its area. In general, the area (A) of a square is side (s) × side (s), A = s².

Note

• Perfect squares are often referred to as square numbers.
• Students should become familiar with the common perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144) and their associated square roots.
• Squares and square roots are inverse operations.

• Rational numbers are any numbers that can be expressed as , where a and b are integers, and b ≠ 0. Examples of rational numbers include: , , −7, 0, 205, 45.328, , −32.5.
• Fractions (positive and negative) are rational numbers. Any fraction can be expressed as a decimal number that either terminates or repeats.
• Whole numbers are rational numbers since any whole number can be expressed as a fraction (e.g., 5 = ).
• Integers (whole numbers and their opposites) are rational numbers since any integer can be expressed as a fraction (e.g., −4 = , +8 = ).
• Rational numbers can be represented as points on a number line to show their relative distance from zero.
• The farther a rational number is to the right of zero on a horizontal number line, the greater the number.
• The farther a rational number is to the left of zero on a horizontal number line, the lesser the number.
• Fractions can be written in a horizontal format (e.g., 1/2 or ½) as well as stacked format (e.g.,).

Note

• There are an infinite number of rational numbers.
• Whole numbers, integers, positive fractions, and positive decimal numbers can be represented using concrete tools.
• Negative fractions and negative decimal numbers can be represented using a number line.
• Negative fractions have the same magnitude as their corresponding positive fraction. The positive and negative signs indicate their relative position to zero. One way of comparing negative fractions is to rewrite them as decimal numbers (e.g.,  = −0.8).
• Negative decimal numbers have the same magnitude as their corresponding positive decimal numbers. The positive and negative signs indicate their relative position to zero.

• Equivalent positive fractions that represent parts of a whole are created by either partitioning or merging partitions.
• A fraction is simplified (in lowest terms) when the numerator (count) and the denominator (unit) have no common whole number factor other than 1 (e.g.,  is in lowest terms,  is not in lowest terms because both the numerator and denominator have a common factor of 2).
• Multiplication and division facts are used to create equivalent fractions and reduce fractions to their lowest terms.
• All unit fractions are in lowest terms.

Note

• Positive and negative fractions can represent quotients. Fractions are equivalent when the results of the numerator divided by the denominator are the same.
• Creating equivalent fractions is used to add and subtract fractions that represent parts of a whole when their units (denominators) are different.
• When performing addition, subtraction, multiplication, and division involving fractions, the results are commonly expressed in lowest terms.
• Sometimes, when working with fractions, a fraction may become a complex fraction in which the numerator and/or the denominator are decimal numbers. To express these fractions in lowest terms, both the numerator and the denominator are multiplied by the appropriate power of ten.

• There are an infinite number of decimal numbers that fall between any two decimal numbers. The place values of the decimal numbers need to be compared to ensure that the number generated does indeed fall between the two numbers.
• The number that falls exactly between any two numbers can be determined by taking the average of the two numbers.
• To determine a fraction between any two fractions, equivalent fractions must be created so that the two fractions have the same denominator in order to do the comparison.

Note

• The number system is infinitely dense. Between any two rational numbers are other rational numbers.

• Rounding makes a number simpler to work with and is often used when estimating computations, measuring, and making quick comparisons.
• A decimal number is rounded to the nearest thousandth, hundredth, tenth, or whole number based on which hundredth, tenth, or whole number it is closest to. If it is the same distance, it is usually rounded up. However, depending on context, it may be rounded down.
• Rounding involves making decisions about what level of precision is needed and is used in measurement, as well as in statistics. How close a rounded number is to the actual amount depends on the unit it is being rounded to: the larger the unit, the broader the estimate; the smaller the unit, the more precise.

Note

• Some decimal numbers do not terminate or repeat. For example, the decimal representation for pi. When working with circles, the decimal representation of pi is usually rounded to the nearest hundredth (3.14).

• Converting between fractions, decimals, and percents often makes calculations and comparisons easier to understand and carry out.
• Relationships of quantities relative to a whole can be expressed as a fraction, a decimal number, and a percent. Percents can be greater than 100%.
• Some fractions can be converted to a percent by creating an equivalent fraction with a denominator of 100.
• 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.
• Some decimal numbers when converted to a percent result in a whole number percent (e.g., 0.6 = 60%, 0.42 = 42%).
• Some decimal numbers when converted to a percent result in a decimal number percent (e.g., 0.423 = 42.3%).
• The relationship of multiplying and dividing whole numbers and decimal numbers by 100 is used to convert between decimal numbers and percents.
• Percents can be understood as decimal hundredths.
• Any percent can be represented as a fraction with a denominator of 100. An equivalent fraction can be created expressed in lowest terms.

Note

• Unit fraction conversions can be scaled to determine non-unit conversions (e.g., one fifth = 0.2, so four fifths is 0.2 × 4 = 0.8). (See  B2.2.)
• Common benchmark fractions, decimals, and percents include:
  •  = 0.5 = 0.50 = 50%
  • = 0.25 = 25%
  • = 0.2 = 0.20 = 20%
  • = 0.125 = 12.5%
  •  = 0.1 = 0.10 = 10%